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