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On a Bernoulli problem with geometric constraints

Antoine Laurain, Yannick Privat (2012)

ESAIM: Control, Optimisation and Calculus of Variations

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

On a Bernoulli problem with geometric constraints

Antoine Laurain, Yannick Privat (2012)

ESAIM: Control, Optimisation and Calculus of Variations

A Bernoulli free boundary problem with geometrical constraints is studied. The domain Ω is constrained to lie in the half space determined by x1 ≥ 0 and its boundary to contain a segment of the hyperplane  {x1 = 0}  where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed...

On semiconvexity properties of rotationally invariant functions in two dimensions

Miroslav Šilhavý (2004)

Czechoslovak Mathematical Journal

Let f be a function defined on the set 𝐌 2 × 2 of all 2 by 2 matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function f can be represented as a function f ˜ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of f in terms of its representation f ˜ .

On the continuity of minimizers for quasilinear functionals

David Cruz-Uribe, Patrizia Di Gironimo, Luigi D'Onofrio (2012)

Czechoslovak Mathematical Journal

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log ( 1 / | x | ) - 1 . Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann....

On the quasiconvex exposed points

Kewei Zhang (2001)

ESAIM: Control, Optimisation and Calculus of Variations

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

On the quasiconvex exposed points

Kewei Zhang (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The notion of quasiconvex exposed points is introduced for compact sets of matrices, motivated from the variational approach to material microstructures. We apply the notion to give geometric descriptions of the quasiconvex extreme points for a compact set. A weak version of Straszewicz type density theorem in convex analysis is established for quasiconvex extreme points. Some examples are examined by using known explicit quasiconvex functions.

On the unique extension problem for functionals of the calculus of variations

Luciano Carbone, Riccardo De Arcangelis (2001)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

By drawing inspiration from the treatment of the non parametric area problem, an abstract functional is considered, defined for every open set in a given class of open subsets of R n and every function in C ∞ ⁢ R n , and verifying suitable assumptions of measure theoretic type, of invariance, convexity, and lower semicontinuity. The problem is discussed of the possibility of extending it, and of the uniqueness of such extension, to a functional verifying analogous properties, but defined in wider families...

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