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We study the quasilinear elliptic problem with multivalued terms.We consider the Dirichlet problem with a multivalued term appearing in the equation and a problem of Neumann type with a multivalued term appearing in the boundary condition. Our approach is based on Szulkin’s critical point theory for lower semicontinuous energy functionals.

Quasilinear elliptic problems with nonstandard growth.

Benboubker, Mohamed Badr, Azroul, Elhoussine, Barbara, Abdelkrim (2011)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Quelques critères pour l’inégalité de Poincaré dans $ℝ^{d}$ , $d \geq 2$

Francis Nier (2002/2003)

Séminaire Équations aux dérivées partielles

Radial minimizer of a $p$ -Ginzburg-Landau type functional with normal impurity inclusion.

Lei, Yutian (2003)

Southwest Journal of Pure and Applied Mathematics [electronic only]

Radial minimizer of a variant of the $p$ -Ginzburg-Landau functional.

Lei, Yutian (2003)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Radial solutions for a nonlocal boundary value problem.

Enguiça, Ricardo, Sanchez, Luís (2006)

Boundary Value Problems [electronic only]

Radially symmetric solutions for a class of critical exponent elliptic problems in $ℝ^{N}$ .

Alves, C.O., de Morais Filho, D.C., Souto, M.A.S. (1996)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Rectifiability of defect measures, fundamental groups and density of Sobolev mappings

Fang Hua Lin (1996)

Journées équations aux dérivées partielles

Regularity of Lipschitz free boundaries for the thin one-phase problem

Daniela De Silva, Ovidiu Savin (2015)

Journal of the European Mathematical Society

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $E (u, Ω) = \int_{Ω} {| \nabla u |}^{2} d X + ℋ^{n} ({u > 0} \cap {x_{n + 1} = 0}), Ω \subset ℝ^{n + 1},$ among all functions $u \geq 0$ which are fixed on $\partial Ω$ .

Regularity of minima: an invitation to the Dark Side of the Calculus of Variations

Giuseppe Mingione (2006)

Applications of Mathematics

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

Regularity of minimizers for a class of membrane energies

Emilio Acerbi, Irene Fonseca, Nicola Fusco (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Regularity of optimal shapes for the Dirichlet’s energy with volume constraint

Tanguy Briancon (2004)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of $ℝ^{d}$ which minimizes the Dirichlet’s energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

Regularity of optimal shapes for the Dirichlet's energy with volume constraint

Tanguy Briancon (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper, we prove some regularity results for the boundary of an open subset of $^{d}$ which minimizes the Dirichlet's energy among all open subsets with prescribed volume. In particular we show that, when the volume constraint is “saturated”, the reduced boundary of the optimal shape (and even the whole boundary in dimension 2) is regular if the state function is nonnegative.

Regularity of solutions fo a degenerate elliptic variational problem.

Luigi Ambrosio (1990)

Manuscripta mathematica

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