This article is concerned with symmetry properties of the solutions of the reaction-diiusion equation u + f (u) = 0 in a bounded connected domain in R N (N = 2; 3; : : :). Of especial interest are nonlinear source terms f of the type f (u) = u p ? u q with 0 q < p 1. Two results are presented. The rst result concerns the solution of a free boundary problem, where the domain is unknown and u and...

In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative ∇k+1u (with k ≥ 1) of a function u to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the k + 1-th directional derivative is about the smallest, while along its perpendicul...

As a follow-up of Haberl-Schuster’s “Asymmetric affine Lp Sobolev inequalities” and Cianchi-Lutwak-Yang-Zhang’s “Affine Moser-Trudinger and Morrey-Sobolev inequalities”, we establish sharp Moser-Trudinger and MorrySobolev inequalities induced by the positive part of a directional derivative on the unit Euclidean sphere. 1. Theorem In their 2009 JFA paper [1], Haberl-Schuster prove the following...

Based on a special concept of the difference of sets, a new notion of the directional derivative of a set-valued map is presented. This theory is applied to set optimization problems with the known set less order relation, and it results in necessary and sufficient optimality conditions.

We calculate the Clarke and Michel-Penot subdifferentials of the function which maps a symmetric matrix to its mth largest eigenvalue. We show these two subdifferentials coincide, and are identical for all choices of index m corresponding to equal eigenvalues. Our approach is via the generalized directional derivatives of the eigenvalue function, thereby completing earlier studies on the classi...

The goal of this paper is two-fold. First we study theoretical properties of discrete spherical means of directional derivatives of a function. Then we focus on the two-dimensional case and use discrete circular means to derive rotation-equivariant discrete approximations of linear and nonlinear firstand second-order differential operators. Applications to nonlinear filtering of digital images ...

In this paper, we are devoted to exploring conditions of well-posedness for generalized hemivariational inequalities with Clarke’s generalized directional derivative in reflexive Banach spaces. By using some equivalent formulations of the generalized hemivariational inequality with Clarke’s generalized directional derivative under different monotonicity assumptions, we establish two kinds of co...

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part Γ of the boundary ∂Ω. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the...

We describe a framework for the design of directional derivative kernels for two-dimensional discrete signals in which we optimize a measure of rotation-equivariance in the Fourier domain. The formulation is applicable to rst-order and higher-order derivatives. We design a set of compact, separable, linear-phase derivative kernels of di erent orders and demonstrate their accuracy.

Here Ω is a bounded domain in RN with smooth boundary, ∆pu = div(|∇u|p−2∇u) is the p-Laplacian, and ∂/∂ν is the outer normal derivative. In the linear case, that is for p = 2, this eigenvalue problem is known as the Steklov problem (see [3]). Problems of the form (1.1) appear in a natural way when one considers the Sobolev trace inequality. In fact, the immersionW1,p(Ω) ↪→ Lp(∂Ω) is compact, he...