This paper addresses the uniqueness and Lipschitz regularity of solutions for a general minimization problem. We assume that the boundary data satisfy the bounded slope condition to ensure the existence of a minimizer, which is globally Lipschitz continuous. With this regularity, we can prove the existence of a unique solution among functions of bounded variation. Finally, we apply these results to a problem related to Hencky plasticity to prove the continuity of the stress and to investigate the regularity and geometry of the level sets of the minimizer.
In this work, we establish a Serrin-type symmetry result for a class of degenerate elliptic operators that naturally arise in the theory of generalized axially symmetric potentials. The relevant differential operator is L-a = partial derivative(rr)u + a/r partial derivative(r) u + Delta(y)u, where a >= 0 represents the fractional parameter.
This paper investigates the existence of solutions for the following class of coupled systems of partial differential equations with an interface condition{ - Delta u i = lambda i a i ( x ) u i + b i ( x ) | u i | p i - 2 u i in Omega i , partial derivative nu u i = gamma ( u 2 - u 1 ) on Sigma , partial derivative eta u 2 = 0 on Gamma , ight.where Sigma = partial derivative Omega 1, Gamma = partial derivative Omega 2 \ Sigma are sign-changing weight functions. The study is divided into different cases according to the values of the exponents p i p_{i} , and the approach is based on variational methods, including the use of the Nehari manifold. Additionally, global existence results for the parabolic counterpart of the system are explored, with an emphasis on the potential well methods.
In this paper we deal with the bounded critical points of a Riesz energy of attractive-repulsive type in dimension 1. Under suitable assumptions on the growth of the kernel in the origin, we are able to prove that they are continuous inside their support.
We consider a variational approach to solve parabolic problems by minimising a functional over time and space. To achieve existence results, we investigate the notion of 𝒜 {\mathscr{A}} -quasiconvexity for non-homogeneous operators in anisotropic spaces. The abstract theory is then applied to formulate a variational solution concept for the generalised Navier–Stokes equations.
