Linear (and nonlinear) Dirichlet problems with singular convection/drift terms
Linear (and nonlinear) Dirichlet
problems with singular
convection/drift terms
LUCIO BOCCARDO
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
This workshop will focus on the most recent developments and achievements in a broad range of topics in the Calculus of Variations, with an emphasis on its applications to material science and imaging. It will provide an exceptional opportunity to present the state of the art of modern methods in the Calculus of Variations and their applications and to stimulate exchange of ideas and knowledge, through a rich selection of talks by leading experts in the field. The conference is an occasion to celebrate the 65th birthday of Gianni Dal Maso and his deep and vast contribution to the Calculus of Variations, from its most fundamental theoretical foundations to its applications to solid mechanics, homogenization, fracture, plasticity, and imaging. It will also be an opportunity to bring together a large community of former students, postdocs, collaborators, and friends who have benefited of Gianni’s knowledge and mathematical advice throughout the years.
Speakers
G. Alberti, Univ. Pisa L. Ambrosio, SNS Pisa L. Boccardo, Sapienza Univ. Roma G. Bouchitté, Univ. Toulon A. Braides, Univ. Roma Tor Vergata G. Buttazzo, Univ. Pisa A. Chambolle*, École Polytechnique, Paris S. Conti, Univ. Bonn G. De Philippis, SISSA and Courant Institute A. De Simone, SISSA and Scuola Superiore Sant’Anna G. Dell’Antonio, Sapienza Univ. Roma and SISSA I. Fonseca, Carnegie Mellon Univ., Pittsburgh G. Francfort, Univ. Paris 13 H. Frankowska, Sorbonne Univ. N. Fusco, Univ. Napoli C. Larsen, Worcester Polytechnic Institute G. Leoni, Carnegie Mellon Univ., Pittsburgh P. Marcellini, Univ. Firenze A. Mielke, WIAS and Humboldt Univ., Berlin J. Morel, ENS Cachan F. Murat, Sorbonne Univ. G. Savare, Univ. Pavia S. Sbordone, Univ. Napoli * to be confirmed
Organisers
V. Chiadò-Piat, Politecnico di Torino A. Garroni, Sapienza Univ. Roma N. Gigli, SISSA M.G. Mora, Univ. Pavia F. Solombrino, Univ. Napoli Federico II R. Toader, Univ. Udine
Where: SISSA
Scuola Internazionale Superiore di Studi Avanzati Via Bonomea 265 Trieste, Italy
When: 27 January – 1 February 2020 For registration: → [email protected] Info: → gianni65.weebly.com
CALCULUS OF VARIATIONS AND APPLICATIONS
Sponsors
AN INTERNATIONAL CONFERENCE TO CELEBRATE GIANNI DAL MASO’S 65th BIRTHDAY
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Thanks for the invitation + excellent organization *
Good morning
Buon giorno
and thanks to the organizers: V. Chiad`o-Piat, A. Garroni, N. Gigli, M.G. Mora, F. Solombrino, R. Toader.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Thanks for the invitation + excellent organization *
Good morning
Buon giorno and thanks to the organizers:
V. Chiad`o-Piat, A. Garroni, N. Gigli, M.G. Mora, F. Solombrino, R. Toader.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Papers concerned with. first part of the talk
L. Boccardo: Some developments on Dirichlet problems with discontinuous coefficients; Boll. Unione Mat. Ital, 2 (2009) 285–297.
(invited paper in memory of 30-death Stampacchia)
L. Boccardo: Dirichlet problems with singular convection terms and applications; J. Differential Equations, 258 (2015) 2290–2314.
L. Boccardo: Stampacchia-Calderon-Zygmund theory for linear elliptic equations with discontinuous coefficients and singular drift; ESAIM, Control, Optimization and Calculus of
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Gianni Dal Maso: good mathematician
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms An International Conference to celebrate Gianni Dal Maso-65
Gianni Dal Maso: good mathematician
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction
Duality
We discuss the existence of distributional solutions for the boundary value problems (the first with a
convection term, the second with a drift term)
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
at least formally, if M(x ) is symmetric, the two above linear problems are in duality.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction
Duality
We discuss the existence of distributional solutions for the boundary value problems (the first with a
convection term, the second with a drift term) ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
at least formally, if M(x ) is symmetric, the two above linear problems are in duality.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction
Duality
We discuss the existence of distributional solutions for the boundary value problems (the first with a
convection term, the second with a drift term) ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
at least formally, if M(x ) is symmetric, the two above linear problems are in duality.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction
Duality
We discuss the existence of distributional solutions for the boundary value problems (the first with a
convection term, the second with a drift term) ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
at least formally, if M(x ) is symmetric, the two above linear problems are in duality.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction Coercivity ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
the differential operators may be not coercive, unless one assumes that either the norm of |E | in LN(Ω) is small, or that div(kE k
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction Coercivity ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
the differential operators may be not coercive, unless one assumes that either the norm of |E | in LN(Ω) is small, or that div(kE k
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Introduction Coercivity ( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, We note that
the differential operators may be not coercive, unless one assumes that either the norm of |E | in LN(Ω) is small, or that div(kE kN) = 0: ...
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Assumptions
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN,
ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1≤ m ≤ ∞,
E ∈ (LN(Ω))N
11: dependence w.r.t. x / 2: nonsmooth dependence /
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Assumptions
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω. Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
11: dependence w.r.t. x / 2: nonsmooth dependence /
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Assumptions
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN,
ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1≤ m ≤ ∞,
E ∈ (LN(Ω))N
11: dependence w.r.t. x / 2: nonsmooth dependence /
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Assumptions
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω.
Ω bounded subset of RN,
ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)|1 ≤ β,
f ∈ Lm(Ω), 1≤ m ≤ ∞,
E ∈ (LN(Ω))N
11: dependence w.r.t. x / 2: nonsmooth dependence /
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Boundary value problem and Lax-Milgram
u weak solution of the boundary value problem −div(M(x)∇u)) = −div(uE (x )) + f (x ) : Ω,
u = 0 : ∂Ω. means u∈ W01,2(Ω) : Z Ω M(x )∇u∇v = Z Ω u E (x )∇v+ Z Ω f (x )v, ∀ v ∈ W01,2(Ω).
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Boundary value problem and Lax-Milgram
u weak solution of the boundary value problem −div(M(x)∇u)) = −div(uE (x )) + f (x ) : Ω,
u = 0 : ∂Ω. means u∈ W01,2(Ω) : Z Ω M(x )∇u∇v = Z Ω u E (x )∇v+ Z Ω f (x )v, ∀ v ∈ W01,2(Ω).
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Coercivity of −div(M(x)∇u) + div(u E (x))
Z Ω M(x )∇v∇v ± Z Ω v E (x )∇v ≥ α Z Ω |∇v|2−h Z Ω |v|2∗i 1 2∗hZ Ω |E (x)|Ni 1 NhZ Ω |∇v|2i 1 2 ≥nα− 1 S hZ Ω|E (x)| Ni 1 NoZ Ω|∇v| 2 E ∈ LN
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Our approach hinges on test function method
The proofs of all the results are very easy
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Setting
Our approach hinges on test function method
The proofs of all the results are very easy
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, 2 ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 3 Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
and we prove for both the b.v.p. the same
Stampacchia-Calderon-Zygmund results of the case
E = 0.
2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, 2 ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 3 Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
and we prove for both the b.v.p. the same
Stampacchia-Calderon-Zygmund results of the case
E = 0.
2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, 2 ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 3 Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
and we prove for both the b.v.p. the same
Stampacchia-Calderon-Zygmund results of the case
E = 0.
2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, 2 ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 3 Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
and we prove for both the b.v.p. the same
Stampacchia-Calderon-Zygmund results of the case
E = 0.
2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Stampacchia-Calderon-Zygmund for the two problems
( −div(M(x)∇u) = −div(u E (x)) + f (x) in Ω, u = 0 on ∂Ω, 2 ( −div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 3 Ω bounded subset of RN, ellipticity: 0< α, α|ξ|2 ≤ M(x)ξξ, |M(x)| ≤ β, f ∈ Lm(Ω), 1≤ m ≤ ∞, E ∈ (LN(Ω))N
and we prove for both the b.v.p. the same
Stampacchia-Calderon-Zygmund results of the case
E = 0.
2 paper invitation U.M.I. in memory of 30-Stampacchia 3 ESAIM-COCV 2019
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W1,q 0 (Ω), q < N−1N . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W1,q 0 (Ω), q < N−1N . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W01,q(Ω), q < N N−1 . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W1,q 0 (Ω), q < N−1N . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W1,q 0 (Ω), q < N−1N . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
|E | ∈ LN(Ω) as E = 0
−div(M(x)∇u)) = −div(u E(x)) + f (x) : Ω,
u = 0 : ∂Ω 1 2N N+2 ≤ m < N 2 ⇒ u ∈W 1,2 0 (Ω)∩ Lm ∗∗ (Ω); 2 1< m < 2N N+2 ⇒ u ∈W 1,m∗ 0 (Ω); 3 m = 1 ⇒ u ∈W1,q 0 (Ω), q < N−1N . u : Z Ω M∇u∇φ = Z Ω u E∇φ + Z Ω fφ, ∀ φ ∈ D. Theorem (70-Brezis) E = 0, m> N 2, it is false that u∈ W 1,m∗ 0 (Ω) Remark E = 0, N+22N +δMeyers < m < N2, u ∈?
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Same results for the drift problem
(
−div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
4
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Same results for the drift problem
(
−div(M(x)∇ψ) = E (x) ∇ψ + g(x) in Ω,
ψ = 0 on ∂Ω,
4
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
Summability properties of solutions
Other recent papers
L. Boccardo, S. Buccheri, G.R. Cirmi: Two linear noncoercive Dirichlet problems in duality; Milan J. Math. 86 (2018), 97–104.
L. Boccardo, S. Buccheri, R.G. Cirmi: Calderon-Zygmund theory for infinite energy solutions of nonlinear elliptic equations with singular drift; preprint.
L. Boccardo, S. Buccheri: A nonlinear homotopy between two linear Dirichlet problems; preprint.
L. Boccardo: Two semilinear Dirichlet problems “almost” in duality; Boll. Unione Mat. Ital. 12 (2019), 349–356.
L. Boccardo, L. Orsina, A. Porretta: Some noncoercive
parabolic equations with lower order terms in divergence form. Dedicated to Philippe B´enilan. J. Evol. Equ. 3 (2003),
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
((E ∈ (L(((
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
((E ∈ (L(((
N(Ω))N
If E 6∈ (LN(Ω))N, even for nothing, as in
|E | ≤ |A||x|, A∈ R , 0 ∈ Ω,
the framework changes completely:
u∈W01,2(Ω) or u∈W01,q(Ω) depends on the size of A . 5 1) if |A| < α(N−2m)m , and N+22N ≤ m < N2, then
u∈ W01,2(Ω)∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m , and 1< m < 2N
N+2, then
u∈ W01,m∗(Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN−1N (Ω))N
and u ∈ W01,q(Ω), for every q< N N−1;
4 if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W01,q(Ω), for
every q< Nα |A|+α
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
((E ∈ (L(((
N(Ω))N
If E 6∈ (LN(Ω))N, even for nothing, as in
|E | ≤ |A||x|, A∈ R , 0 ∈ Ω,
the framework changes completely:
u∈W01,2(Ω) or u∈W01,q(Ω) depends on the size of A . 5
1) if |A| < α(N−2m)m , and N+22N ≤ m < N2, then u∈ W01,2(Ω)∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m , and 1< m < 2N
N+2, then
u∈ W01,m∗(Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN−1N (Ω))N
and u ∈ W01,q(Ω), for every q< N N−1;
4 if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W01,q(Ω), for
every q< Nα |A|+α
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
((E ∈ (L(((
N(Ω))N
If E 6∈ (LN(Ω))N, even for nothing, as in
|E | ≤ |A||x|, A∈ R , 0 ∈ Ω,
the framework changes completely:
u∈W01,2(Ω) or u∈W01,q(Ω) depends on the size of A . 5 1) if |A| < α(N−2m)m , and N+22N ≤ m < N2, then
u∈ W01,2(Ω)∩ Lm∗∗(Ω);
2) if |A| < α(N−2m)m , and 1< m < 2N
N+2, then
u∈ W01,m∗(Ω);
3) if |A| < α(N − 2), and m = 1, then ∇u ∈ (MN−1N (Ω))N
and u ∈ W01,q(Ω), for every q< N N−1;
4 if α(N − 2) ≤ |A| < α(N − 1), then u ∈ W01,q(Ω), for
every q< Nα |A|+α
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
((E ∈ (L(((
N(Ω))N Radial ex.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L((( 2(Ω))N E ∈ (L2(Ω))N ( definition of solution; existence of solution. 6 6JDE 2015
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L((( 2(Ω))N E ∈ (L2(Ω))N ( definition of solution; existence of solution. 6 6JDE 2015
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + n1|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f | ! only (again) E ∈ L2 is needed.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + 1n|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f | ! only (again) E ∈ L2 is needed.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + 1n|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f |
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + 1n|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f | !
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + 1n|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f | ! only (again) E ∈ L2 is needed.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
If we add the zero order term “+u”, the framework changes completely.
A , un∈ W01,2(Ω) : −div(M(x)∇un) + A un =−div un 1 + 1n|un| E (x )+ f (x ) Third estimate A Z Ω |un| ≤ Z Ω |f | ! only (again) E ∈ L2 is needed.
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
By duality: problems with very singular drifts
( −div(M(x)∇ψ) + ψ = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 7 E ∈ L28, f bounded ⇒ u ∈ W1,2 0 (Ω), bounded. 7 DIE 2019 8 only L2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
By duality: problems with very singular drifts
( −div(M(x)∇ψ) + ψ = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 7 E ∈ L28, f bounded ⇒ u ∈ W1,2 0 (Ω), bounded. 7 DIE 2019 8 only L2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
By duality: problems with very singular drifts
( −div(M(x)∇ψ) + ψ = E (x) ∇ψ + g(x) in Ω, ψ = 0 on ∂Ω, 7 E ∈ L28, f bounded ⇒ u ∈ W1,2 0 (Ω), bounded. 7 DIE 2019 8 only L2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
An elliptic system connected with the mathematical study of PDE models for chemotaxis −div(A(x)∇u) + u =−div(uM(x )∇ψ) + f (x ), −div(M(x)∇ψ) = uθ. 910 9JDE 2015 10Comm.PDE + L. Orsina
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms Existence of weak/distributional solutions
(((( (((( ((E ∈ (L(((
2(Ω))N
An elliptic system connected with the mathematical study of PDE models for chemotaxis −div(A(x)∇u) + u =−div(uM(x )∇ψ) + f (x ), −div(M(x)∇ψ) = uθ. 910 9JDE 2015 10Comm.PDE + L. Orsina
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
The starting point
((((
((((
kE k
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
The starting point
Nonlinearapproach tolinear problems
un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| E (x )+f (x ) that is un∈ W01,2(Ω) : Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, ∀ v ∈ W01,2(Ω). Existence un: Schauder
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
The starting point
Nonlinearapproach tolinear problems
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x )+f (x ) that is un∈ W01,2(Ω) : Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, ∀ v ∈ W01,2(Ω). Existence un: Schauder
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
The starting point
Nonlinearapproach tolinear problems
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x )+f (x ) that is un∈ W01,2(Ω) : Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, ∀ v ∈ W01,2(Ω). Existence un: Schauder
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
The starting point
Nonlinearapproach tolinear problems
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x )+f (x ) that is un∈ W01,2(Ω) : Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, ∀ v ∈ W01,2(Ω). Existence un: Schauder
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Log-estimate (first estimate)
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x ) +f (x ) test f. un 1 +|un|⇒α Z Ω |∇un|2 (1 +|un|)2≤ Z Ω |un| 1 +|un||E | |∇un| (1 +|un|) + Z Ω |fn| α 2 Z Ω |∇un|2 (1 +|un|)2 ≤ 1 2α Z Ω|E | 2+ Z Ω|f |, Z Ω |∇ log(1 + |un|)|2 ≤ 1 α2 Z Ω |E |2+ 2 α Z Ω |f | hZ Ω | log(1 + |un|)|2 ∗i2∗2 ≤ 1 S2α2 Z Ω |E |2+ 2 S2α Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Log-estimate (first estimate)
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x ) +f (x ) test f. un 1 +|un|⇒α Z Ω |∇un|2 (1 +|un|)2≤ Z Ω |un| 1 +|un||E | |∇un| (1 +|un|) + Z Ω |fn| α 2 Z Ω |∇un|2 (1 +|un|)2 ≤ 1 2α Z Ω|E | 2+ Z Ω|f |, Z Ω |∇ log(1 + |un|)|2 ≤ 1 α2 Z Ω |E |2+ 2 α Z Ω |f | hZ Ω | log(1 + |un|)|2 ∗i2∗2 ≤ 1 S2α2 Z Ω |E |2+ 2 S2α Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Log-estimate (first estimate)
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x ) +f (x ) test f. un 1 +|un|⇒α Z Ω |∇un|2 (1 +|un|)2≤ Z Ω |un| 1 +|un||E | |∇un| (1 +|un|) + Z Ω |fn| α 2 Z Ω |∇un|2 (1 +|un|)2 ≤ 1 2α Z Ω|E | 2+ Z Ω|f |, Z Ω |∇ log(1 + |un|)|2 ≤ 1 α2 Z Ω |E |2+ 2 α Z Ω |f | hZ Ω | log(1 + |un|)|2 ∗i2∗2 ≤ 1 S2α2 Z Ω |E |2+ 2 S2α Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Log-estimate (first estimate)
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x ) +f (x ) test f. un 1 +|un|⇒α Z Ω |∇un|2 (1 +|un|)2≤ Z Ω |un| 1 +|un||E | |∇un| (1 +|un|) + Z Ω |fn| α 2 Z Ω |∇un|2 (1 +|un|)2 ≤ 1 2α Z Ω|E | 2+ Z Ω|f |, Z Ω |∇ log(1 + |un|)|2 ≤ 1 α2 Z Ω |E |2+ 2 α Z Ω |f | hZ Ω | log(1 + |un|)|2 ∗i2∗2 ≤ 1 S2α2 Z Ω |E |2+ 2 S2α Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Log-estimate (first estimate)
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1 n|un| E (x ) +f (x ) test f. un 1 +|un|⇒α Z Ω |∇un|2 (1 +|un|)2≤ Z Ω |un| 1 +|un||E | |∇un| (1 +|un|) + Z Ω |fn| α 2 Z Ω |∇un|2 (1 +|un|)2 ≤ 1 2α Z Ω|E | 2+ Z Ω|f |, Z Ω |∇ log(1 + |un|)|2 ≤ 1 α2 Z Ω |E |2+ 2 α Z Ω |f | hZ Ω | log(1 + |un|)|2 ∗i2∗2 ≤ 1 S2α2 Z Ω |E |2+ 2 S2α Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
Tk(s) −k −k k k −k k Gk(s) −k k Th(Gk(s))/h −k − h -1 k + h 1 1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
Tk(s) −k −k k k −k k Gk(s) −k k Th(Gk(s))/h −k − h -1 k + h 1 1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
un∈W01,2(Ω) :−div(M(x)∇un) =−div un 1 + n1|un| E (x )+f α Z Ω |∇Tk(un)|2 ≤ Z Ω |un| |E ||∇Tk(un)|+ Z Ω fTk(un) ≤ k Z Ω |E ||∇Tk(un)| + k Z Ω |f | ≤ α 2 Z Ω |∇Tk(un)|2+ 2 αk 2 Z Ω |E |2 + k Z Ω |f |⇒ α 2 Z Ω |∇Tk(un)|2 ≤ 2 αk 2 Z Ω |E |2+ k Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
un∈W01,2(Ω) :−div(M(x)∇un) =−div un 1 + n1|un| E (x )+f α Z Ω |∇Tk(un)|2 ≤ Z Ω |un| |E ||∇Tk(un)|+ Z Ω fTk(un) ≤ k Z Ω |E ||∇Tk(un)| + k Z Ω |f | ≤ α 2 Z Ω |∇Tk(un)|2+ 2 αk 2 Z Ω |E |2 + k Z Ω |f |⇒ α 2 Z Ω |∇Tk(un)|2 ≤ 2 αk 2 Z Ω |E |2+ k Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
un∈W01,2(Ω) :−div(M(x)∇un) =−div un 1 + n1|un| E (x )+f α Z Ω |∇Tk(un)|2 ≤ Z Ω |un| |E ||∇Tk(un)|+ Z Ω fTk(un) ≤ k Z Ω |E ||∇Tk(un)| + k Z Ω |f | ≤ α2 Z Ω|∇T k(un)|2+ 2 αk 2 Z Ω|E | 2 + k Z Ω|f |⇒ α 2 Z Ω |∇Tk(un)|2 ≤ 2 αk 2 Z Ω |E |2+ k Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Truncation-estimate (second estimate)
un∈W01,2(Ω) :−div(M(x)∇un) =−div un 1 + n1|un| E (x )+f α Z Ω |∇Tk(un)|2 ≤ Z Ω |un| |E ||∇Tk(un)|+ Z Ω fTk(un) ≤ k Z Ω |E ||∇Tk(un)| + k Z Ω |f | ≤ α2 Z Ω|∇T k(un)|2+ 2 αk 2 Z Ω|E | 2 + k Z Ω|f |⇒ α 2 Z Ω |∇Tk(un)|2 ≤ 2 αk 2 Z Ω |E |2+ k Z Ω |f |1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Two important estimates
un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1n|un| E (x )+f (x ) hZ Ω| log(1 + |u n|)|2 ∗i2∗2 ≤ S21α2 Z Ω|E | 2+ 2 S2α Z Ω|f | 1 α 2 Z Ω|∇T k(un)|2 ≤ 2 αk 2 Z Ω|E | 2+ k Z Ω|f | 1
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates kGk(un)k W01,2(Ω)≤ C (E , f , α) Proof of kGk(un)kW1,2 0 (Ω) ≤ C (E , f , α), k > kE(kE kN)
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates
Let δ > 0. Use Gk(un) as test f.; Young, H¨older, Sob.⇒
α Z Ω |∇Gk(un)|2≤ Z Ω |Gk(un)||E ||∇Gk(un)|+k Z Ω |E ||∇Gk(un)|+ Z Ω |Gk(un)||f | ≤ 1 S( Z k<|un| |E |N)N1 Z Ω|∇G k(un)|2+δ Z Ω|∇G k(un)|2+ k2 4δ Z k<|un| |E |2 +δ Z Ω |∇Gk(un)|2+ S 2 4δ Z k<|un| |f |N+22N N+2N , " α− 1 S Z k<|un| |E |N N1 − 2δ # Z Ω |∇Gk(un)|2 ≤ k 2 4δ Z k<|un| |E |2+S 2 4δ Z k<|un| |f |N+22N N+2N .
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates " α− 1 S Z k<|un| |E |N N1 − 2δ # Z Ω |∇Gk(un)|2 ≤ k 2 4δ Z k<|un| |E |2 +S 2 4δ Z k<|un| |f |N+22N N+2N .
Fix δ so that 2δ = α4. Then log-estimate implies that there exists kE, such that
1 S Z k<|un| |E |N N1 ≤ α 4, k ≥ kE. Thus, for some C1 > 0, we have, if k ≥ kE,
C (kE k LN) Z Ω|∇G k(un)|2 ≤ C1k2 Z k<|un| |E |2+C 2 Z k<|un| |f |N+22N N+2N .
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates 3 estimates hZ Ω| log(1 + |u n|)|2 ∗i2∗2 ≤ S21α2 Z Ω|E | 2+ 2 S2α Z Ω|f | 1 α 2 Z Ω|∇T k(un)|2 ≤ 2 αk 2 Z Ω|E | 2+ k Z Ω|f | 1, forall k > 0 C (kE k LN) Z Ω|∇G k(un)|2≤k2 Z k<|un| |E |2+ Z k<|un| |f |N+22N N+2N , k≥k(kE k N)
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates k = kE C0 Z Ω |∇Tk(un)|2 ≤ k2 Z Ω |E |2+ k Z Ω |f |1, forall k > 0 C (kE k LN) Z Ω |∇Gk(un)|2≤k2 Z k<|un| |E |2+ Z k<|un| |f |N+22N N+2N , k≥k(kE kN) h C0+C (kE kLN) iZ Ω |∇un|2 ≤ 2k2 Z Ω |E |2+k Z Ω |f |1+ Z Ω |f |N+22N N+2N L([tun]) =−div([tun] E ) + (tf ) t2hC0+C (kE kLN) iZ Ω |∇un|2≤2k2 Z Ω |E |2 +tk Z Ω |f |1 +t2 Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates k = kE C0 Z Ω |∇Tk(un)|2 ≤ k2 Z Ω |E |2+ k Z Ω |f |1, forall k > 0 C (kE k LN) Z Ω |∇Gk(un)|2≤k2 Z k<|un| |E |2+ Z k<|un| |f |N+22N N+2N , k≥k(kE kN) h C0+C (kE kLN) iZ Ω |∇un|2 ≤ 2k2 Z Ω |E |2+k Z Ω |f |1+ Z Ω |f |N+22N N+2N L([tun]) =−div([tun] E ) + (tf ) t2hC0+C (kE kLN) iZ Ω |∇un|2≤2k2 Z Ω |E |2 +tk Z Ω |f |1 +t2 Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates k = kE C0 Z Ω |∇Tk(un)|2 ≤ k2 Z Ω |E |2+ k Z Ω |f |1, forall k > 0 C (kE k LN) Z Ω |∇Gk(un)|2≤k2 Z k<|un| |E |2+ Z k<|un| |f |N+22N N+2N , k≥k(kE kN) h C0+C (kE kLN) iZ Ω |∇un|2 ≤ 2k2 Z Ω |E |2+k Z Ω |f |1+ Z Ω |f |N+22N N+2N L([tun]) =−div([tun] E ) + (tf ) t2hC0+C (kE kLN) iZ Ω |∇un|2≤2k2 Z Ω |E |2 +tk Z Ω |f |1 +t2 Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates k = kE C0 Z Ω |∇Tk(un)|2 ≤ k2 Z Ω |E |2+ k Z Ω |f |1, forall k > 0 C (kE k LN) Z Ω |∇Gk(un)|2≤k2 Z k<|un| |E |2+ Z k<|un| |f |N+22N N+2N , k≥k(kE kN) h C0+C (kE kLN) iZ Ω |∇un|2 ≤ 2k2 Z Ω |E |2+k Z Ω |f |1+ Z Ω |f |N+22N N+2N L([tun]) =−div([tun] E ) + (tf ) t2hC0+C (kE kLN) iZ Ω |∇un|2≤2k2 Z Ω |E |2 +tk Z Ω |f |1 +t2 Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates t2hC0+C (kE kLN) iZ Ω |∇un|2≤2kE2 Z Ω |E |2+tk E Z Ω |f |1+t2 Z Ω |f |N+22N N+2N h C0+C (kE kLN) iZ Ω |∇un|2≤ 2k2 E t2 Z Ω |E |2+kE t Z Ω |f |1+ Z Ω |f |N+22N N+2N lim t→∞ h C0+ C (kE kLN) iZ Ω |∇un|2 ≤ Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates t2hC0+C (kE kLN) iZ Ω |∇un|2≤2kE2 Z Ω |E |2+tk E Z Ω |f |1+t2 Z Ω |f |N+22N N+2N h C0+C (kE kLN) iZ Ω |∇un|2≤ 2k2 E t2 Z Ω |E |2+kE t Z Ω |f |1+ Z Ω |f |N+22N N+2N lim t→∞ h C0+ C (kE kLN) iZ Ω |∇un|2 ≤ Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates t2hC0+C (kE kLN) iZ Ω |∇un|2≤2kE2 Z Ω |E |2+tk E Z Ω |f |1+t2 Z Ω |f |N+22N N+2N h C0+C (kE kLN) iZ Ω |∇un|2≤ 2k2 E t2 Z Ω |E |2+kE t Z Ω |f |1+ Z Ω |f |N+22N N+2N lim t→∞ h C0+ C (kE kLN) iZ Ω |∇un|2 ≤ Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates t2hC0+C (kE kLN) iZ Ω |∇un|2≤2kE2 Z Ω |E |2+tk E Z Ω |f |1+t2 Z Ω |f |N+22N N+2N h C0+C (kE kLN) iZ Ω |∇un|2≤ 2k2 E t2 Z Ω |E |2+kE t Z Ω |f |1+ Z Ω |f |N+22N N+2N lim t→∞ h C0+ C (kE kLN) iZ Ω |∇un|2 ≤ Z Ω |f |N+22N N+2N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates un∈ W01,2(Ω) :−div(M(x)∇un) =−div un 1 + 1n|un| E (x )+f (x ) r h C0+ C (kE kLN) i Z Ω|∇u n|2 12 ≤ kf kL 2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω).
Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, Passing to the limit as n tends to infinity yields the result. Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω). Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, Passing to the limit as n tends to infinity yields the result. Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω). Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v,
Passing to the limit as n tends to infinity yields the result. Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω). Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, Passing to the limit as n tends to infinity yields the result.
Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω). Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, Passing to the limit as n tends to infinity yields the result. Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms The convection problem
Estimates Theorem
Let E ∈ (LN(Ω))N, f ∈ Lm(Ω), m≥ 2N
N+2. Then there exists
u∈ W01,2(Ω) weak solution, that is Z Ω M(x )∇u∇v = Z Ω u E (x )∇v + Z Ω f v, ∀ v ∈ W01,2(Ω). Since the sequence {un} is bounded in W01,2(Ω), then,
up to subsequences, un converges weakly in W01,2(Ω) to
a function u. Z Ω M(x )∇un∇v = Z Ω un 1 + 1n|un| E (x )∇v + Z Ω f (x )v, Passing to the limit as n tends to infinity yields the result. Moreover (thanks to the weak lower s.c.)
q C0+ C (kE kLN) hR Ω|∇u| 2i 1 2 ≤ kf kL2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Last part of the talk: new presentation of some results
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Last part of the talk: new presentation of some results
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Estimate on the sequence {un}
r h C0+ C (kE kLN) i Z Ω |∇un|2 12 ≤ kf kL 2N N+2 un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| 1 1 + 1n|∇ψn| E (x )+ f (x ) Z Ω M∗(x )∇ψn∇un= Z Ω E (x )·∇ψn 1 1 + n1|∇ψn| 1 1 + n1|un| un+ Z Ω f (x )ψn ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf kN+22N kgkN+22N ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Estimate on the sequence {un}
r h C0+ C (kE kLN) i Z Ω |∇un|2 12 ≤ kf kL 2N N+2 un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| 1 1 + 1n|∇ψn| E (x )+ f (x ) Z Ω M∗(x )∇ψn∇un= Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + n1|un| un+ Z Ω f (x )ψn ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf kN+22N kgkN+22N ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Estimate on the sequence {un}
r h C0+ C (kE kLN) i Z Ω |∇un|2 12 ≤ kf kL 2N N+2 un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| 1 1 + 1n|∇ψn| E (x )+ f (x ) Z Ω M∗(x )∇ψn∇un= Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + n1|un| un+ Z Ω f (x )ψn ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf kN+22N kgkN+22N ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Estimate on the sequence {un}
r h C0+ C (kE kLN) i Z Ω |∇un|2 12 ≤ kf kL 2N N+2 un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| 1 1 + 1n|∇ψn| E (x )+ f (x ) Z Ω M∗(x )∇ψn∇un= Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + n1|un| un+ Z Ω f (x )ψn ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf kN+22N kgkN+22N ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality
Drift problem
Estimate on the sequence {un}
r h C0+ C (kE kLN) i Z Ω |∇un|2 12 ≤ kf kL 2N N+2 un∈ W01,2(Ω) : −div(M(x)∇un) =−div un 1 + 1n|un| 1 1 + 1n|∇ψn| E (x )+ f (x ) Z Ω M∗(x )∇ψn∇un= Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + n1|un| un+ Z Ω f (x )ψn ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf kN+22N kgkN+22N ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗(x ) ∇ψn) = E (x )· ∇ψn 1 + 1 n|∇ψn| 1 1 + 1 n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf k 2N N+2kgk 2N N+2 ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2 Z Ω M∗(x )∇ψn∇ψn≤ kE kLN Z Ω |∇ψn|2 12 ˜ C (kE k N)kgkN+22N + ˜C (kE kN)kgk 2 2N N+2 ⇒ α2 Z Ω|∇ψ n|2 ≤ 1 2αkE k 2 LN ˜ C (kE k N) 2 kgk2 2N N+2 + ˜C (kE k N)kgk 2 2N N+2 : Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgkN+22N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗(x ) ∇ψn) = E (x )· ∇ψn 1 + 1 n|∇ψn| 1 1 + 1 n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf k2N N+2kgk 2N N+2 ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2 Z Ω M∗(x )∇ψn∇ψn≤ kE kLN Z Ω |∇ψn|2 12 ˜ C (kE k N)kgkN+22N + ˜C (kE kN)kgk 2 2N N+2 ⇒ α2 Z Ω|∇ψ n|2 ≤ 1 2αkE k 2 LN ˜ C (kE k N) 2 kgk2 2N N+2 + ˜C (kE k N)kgk 2 2N N+2 : Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgkN+22N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗(x ) ∇ψn) = E (x )· ∇ψn 1 + 1 n|∇ψn| 1 1 + 1 n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf k2N N+2kgk 2N N+2 ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2 Z Ω M∗(x )∇ψn∇ψn≤ kE kLN Z Ω |∇ψn|2 12 ˜ C (kE k N)kgkN+22N + ˜C (kE kN)kgk 2 2N N+2 ⇒ α2 Z Ω|∇ψ n|2 ≤ 1 2αkE k 2 LN ˜ C (kE k N) 2 kgk2 2N N+2 + ˜C (kE k N)kgk 2 2N N+2 : Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgkN+22N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗(x ) ∇ψn) = E (x )· ∇ψn 1 + 1 n|∇ψn| 1 1 + 1 n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf k2N N+2kgk 2N N+2 ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2 Z Ω M∗(x )∇ψn∇ψn≤ kE kLN Z Ω |∇ψn|2 12 ˜ C (kE k N)kgkN+22N + ˜C (kE kN)kgk 2 2N N+2 ⇒ α2 Z Ω|∇ψ n|2 ≤ 1 2αkE k 2 LN ˜ C (kE k N) 2 kgk2 2N N+2 + ˜C (kE k N)kgk 2 2N N+2 : Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgkN+22N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗(x ) ∇ψn) = E (x )· ∇ψn 1 + 1 n|∇ψn| 1 1 + 1 n|un| + g (x ) Z Ω ψnf = Z Ω ung ≤ kunk2∗kgk2N N+2 ≤ ˜ C (kE k N)kf k2N N+2kgk 2N N+2 ⇒ kψnk2∗ ≤ ˜C (kE kN)kgk2N N+2 Z Ω M∗(x )∇ψn∇ψn≤ kE kLN Z Ω |∇ψn|2 12 ˜ C (kE k N)kgkN+22N + ˜C (kE kN)kgk 2 2N N+2 ⇒ α2 Z Ω|∇ψ n|2 ≤ 1 2αkE k 2 LN ˜ C (kE k N) 2 kgk2 2N N+2 + ˜C (kE k N)kgk 2 2N N+2 : Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgkN+22N
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgk2N N+2 ∇ψn *∇ ˜ψ;
now it is not enough. Note: un ∈ W01,2(Ω) :−div(M
∗(x )
∇ψn) = yn* in L
2N N+2(Ω)
It is possible to prove that11 ∇ψ
n(x )→ ∇ ˜ψ(x), a.e. Z Ω M∗(x )∇ψn∇v = Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + 1n|un| v + Z Ω f v
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgk2N N+2 ∇ψn*∇ ˜ψ;
now it is not enough. Note: un ∈ W01,2(Ω) :−div(M
∗(x )
∇ψn) = yn* in L
2N N+2(Ω)
It is possible to prove that11 ∇ψ
n(x )→ ∇ ˜ψ(x), a.e. Z Ω M∗(x )∇ψn∇v = Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + 1n|un| v + Z Ω f v
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgk2N N+2 ∇ψn*∇ ˜ψ;
now it is not enough.
Note: un ∈ W01,2(Ω) :−div(M ∗(x )
∇ψn) = yn* in L
2N N+2(Ω)
It is possible to prove that11 ∇ψ
n(x )→ ∇ ˜ψ(x), a.e. Z Ω M∗(x )∇ψn∇v = Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + 1n|un| v + Z Ω f v
Linear (and nonlinear) Dirichlet problems with singular convection/drift terms By duality Drift problem ψn ∈ W01,2(Ω) :−div(M ∗ (x )∇ψn) = E (x )· ∇ψn 1 + 1n|∇ψn| 1 1 + 1n|un| + g (x ) Z Ω |∇ψn|2 12 ≤ ¯C (kE k N)kgk2N N+2 ∇ψn*∇ ˜ψ;
now it is not enough. Note: un ∈ W01,2(Ω) :−div(M
∗(x )
∇ψn) = yn* in L
2N N+2(Ω)
It is possible to prove that11 ∇ψ
n(x )→ ∇ ˜ψ(x), a.e. Z Ω M∗(x )∇ψn∇v = Z Ω E (x )·∇ψn 1 1 + 1n|∇ψn| 1 1 + 1n|un| v + Z Ω f v
