Related papers: Gradient estimates in fractional Dirichlet problem…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these…
This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…
This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling H\"older regularity…
This paper surveys recent analytical and numerical research on linear problems for the integral fractional Laplacian, fractional obstacle problems, and fractional minimal graphs. The emphasis is on the interplay between regularity,…
We study the Dirichlet problem of a class of fully nonlinear elliptic equations on Hermitian manifolds and derive a priori $C^2$ estimates which depend on the initial data on manifolds, the admissible subsolutions and the upper bound of the…
This paper presents necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a Lagrangian depending on the free end-points. The fractional derivatives are defined in the sense of Caputo.
This paper is concerned with the derivation of conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains. These estimates provide computable and guaranteed upper and lower…
We provide sharp boundary regularity estimates for solutions to elliptic equations driven by an integro-differential operator obtained as the sum of a Laplacian with a nonlocal operator generalizing a fractional Laplacian. Our approach…
The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting…
Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on…
In this paper, we develop a series of boundary pointwise regularity for Dirichlet problems and oblique derivative problems. As applications, we give direct and simple proofs of the higher regularity of the free boundaries in obstacle-type…
In the paper, we derive an existence result for a nonlinear nonautonomous partial elliptic system on an open bounded domain with Dirichlet boundary conditions, containg fractional powers of the weak Dirichlet-Laplace operator that are meant…
We prove a Yau's type gradient estimate for positive $f$-harmonic functions with the Dirichlet boundary condition on smooth metric measure spaces with compact boundary when the infinite dimensional Bakry-Emery Ricci tensor and the weighted…
The present paper studies the fractional $p$-Laplacian boundary value problems with jumping nonlinearities at zero or infinity and obtain the existence of multiple solutions and sign-changing solutions by constructing the suitable…
In a previous paper we considered a class of infinitely degenerate quasilinear equations and derived a priori bounds for high order derivatives of solutions in terms of the Lipschitz norm. We now show that it is possible to obtain bounds…
A common approach is present concerning the problem of Dirichlet, both for bounded 3D domains and their (unbounded) complements, regarding the fractional (3D) Poisson equation.
We show that arbitrary homeomorphic solutions to the Beltrami equations with generalized derivatives satisfy certain moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions and prove a series of…
In this paper, we consider the approximate controllability of partial differential equations with time derivatives of non-integer order via boundary control. We first show the unique existence of the solution under smooth boundary…
The mixed problem for a degenerate high order equation with a fractional derivative in a rectangular domain is considered in the article. The existence of a solution and its uniqueness are shown by the spectral method.