1,720,991 research outputs found
Functions with prescribed singular values of the gradient.
No full textWe prove the existence of infinitely many vector-valued Lipschitz-continuous functions u on an open set Ω satisfying suitable Dirichlet boundary conditions such that the singular values of the gradient matrix ∇u, agree a.e. on Ω with N given positive, bounded and lower semicontinuous functions
Existence of solutions for a class of non convex minimum problems
No full textIn this paper we give sufficient conditions for the existence of solutions to the problem of minimizing the integral of [f ( ∇v) + v] on a convex n-dimensional set Ω . Here f is nonnegative, nonconvex, Borel-measurable, and vanishes on the boundary of a convex n-dimensional set K
On the minimum problem for nonconvex, multiple integrals of product type
No full textWe consider the problem of minimizing multiple integrals of product type, i.e. (P) min [GRAPHICS] where Omega is a bounded, open set in R-N, f: R-N --> [0, infinity) is a possibly nonconvex, lower semicontinuous function with p-growth at infinity for some 1 R is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that the class of coefficents g that yield existence to (P) is dense in the space of continuous, positive functions on R. We present examples which show that these conditions for attainment are essentially sharp
Minimizing nonconvex, simple integrals of product type
No full textWe consider the problem of minimizing simple integrals of product type, i.e. min {integral (T)(0) g(x(t))f(x ́ (t)) dt: x is an element of AC([0, T]), x(0) = x(0), x(T) = x(T)}. where f:R --> [0, proportional to] is a possibly nonconvex, lower semicontinuous function with either superlinear or slow growth at infinity. Assuming that the relaxed problem (P**) obtained from (P) by replacing f with its convex envelope f** admits a solution. we prove attainment for (P) for every continuous, positively bounded below the coefficient g such that (i) every point t is an element ofR is squeezed between two intervals where g is monotone and (ii) g has no strict local minima. This shows in particular that, for those f such that the relaxed problem (P**) has a solution, the class of coefficients g that yield existence to (P) is dense in the space of continuous, positive Functions on R. We discuss various instances of growth conditions on f that yield solutions to (P**) and we present examples that show that the hypotheses on g considered above for attainment are essentially sharp
Vectorial Hamilton-Jacobi equations with rank one affine dependence on the gradient.
No full textThis paper deals with Dirichlet problems for vectorial, stationary Hamilton-Jacobi equation
Minimizing non-convex multiple integrals: a density result.
No full textWe consider variational problems whose lagrangian is of the form f(Du)+g(u) where f is a possibly non-convex lower semicontinuous function with p-growth at infinity for some 1 < p < ∞, and the boundary datum is any function in W 1,p (Ω). Assuming that the convex envelope of f is affine on each connected component of the set {f ^∗∗ < f }, we prove the existence of solutions to (P) for every continuous function g such that (i) g has no strict local minima and (ii) every convergent sequence of extremum points of g eventually belongs to an interval where g is constant, thus showing that the set of continuous functions g that yield existence to (P) is dense in the space of continuous functions on R
On a class of nonconvex Bolza problems related to Blatz-Ko elastic materials
No full textWe study the existence of solutions to Bolza problems for a special class of one-dimensional, nonconvex integrals. These integrals describe the possibly singular, radial deformations of certain rubberlike materials called Blatz–Ko materials
Existence of minimizers for nonconvex, noncoercive simple integrals.
No full textWe consider the problem of minimizing autonomous, simple integrals such as \min\,\left\{ \int_0^T f\left(x(t)\,,x^\prime(t)\right)\,dt\colon\,\, \text{, , } \right\}, \tag{} where is a possibly nonconvex function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ()---obtained from () by replacing f with its convex envelope f** with respect to the derivative variable ---admits a solution, we prove attainment for () under mild regularity and growth assumptions on f and f**. We discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem (), and we present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp
Polyconvex energies and cavitation
Get PDFWe study the existence of singular minimizers in the class of radial deformations for polyconvex energies that grow linearly with respect to the Jacobian
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