MDPI - Publisher of Open Access Journals

9 pages, 276 KB

Open AccessArticle

Differential-Difference Elliptic Equations with Nonlocal Potentials in Half-Spaces

by Andrey B. Muravnik

Mathematics 2023, 11(12), 2698; https://doi.org/10.3390/math11122698 - 14 Jun 2023

Cited by 5 | Viewed by 1186

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity. Full article

(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms)

22 pages, 377 KB

Open AccessArticle

Schrödinger Harmonic Functions with Morrey Traces on Dirichlet Metric Measure Spaces

by Tianjun Shen and Bo Li

Mathematics 2022, 10(7), 1112; https://doi.org/10.3390/math10071112 - 30 Mar 2022

Cited by 1 | Viewed by 1931

Abstract

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

[...] Read more.

Assume that

(X, d, μ)

is a metric measure space that satisfies a Q-doubling condition with

Q > 1

and supports an

L^{2}

-Poincaré inequality. Let

𝓛

be a nonnegative operator generalized by a Dirichlet form

E

and V be a Muckenhoupt weight belonging to a reverse Hölder class

R H_{q} (X)

for some

q \geq (Q + 1) / 2

. In this paper, we consider the Dirichlet problem for the Schrödinger equation

- \partial_{t}^{2} u + 𝓛 u + V u = 0

on the upper half-space

X \times R_{+}

, which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space

R^{Q}

to the metric measure space X and improves the reverse Hölder index from

q \geq Q

q \geq (Q + 1) / 2

. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

37 pages, 479 KB

Open AccessArticle

On the Global Well-Posedness and Decay of a Free Boundary Problem of the Navier–Stokes Equation in Unbounded Domains

by Kenta Oishi and Yoshihiro Shibata

Mathematics 2022, 10(5), 774; https://doi.org/10.3390/math10050774 - 28 Feb 2022

Cited by 2 | Viewed by 2220

Abstract

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in

L_{p}

in time and

L_{q}

in space framework in a uniformly

H_{\infty}^{2}

[...] Read more.

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in

L_{p}

in time and

L_{q}

in space framework in a uniformly

H_{\infty}^{2}

domain

Ω \subset R^{N}

for

N \geq 4

. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the

L_{q}

L_{r}

estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction

N \geq 4

is required to deduce an estimate for the nonlinear term

G (u)

arising from

{div}_{} v = 0

. However, we establish the results in the half space

R_{+}^{N}

for

N \geq 3

by reducing the linearized problem to the problem with

G = 0

, where

G

is the right member corresponding to

G (u)

. Full article

(This article belongs to the Special Issue Maximal Regularity, Stability Estimates and Mathematical Fluid Dynamics II)

16 pages, 856 KB

Open AccessArticle

Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

by Jean-Daniel Djida and Arran Fernandez

Axioms 2018, 7(3), 65; https://doi.org/10.3390/axioms7030065 - 1 Sep 2018

Cited by 3 | Viewed by 4375

Abstract

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation [...] Read more.

(This article belongs to the Special Issue Mathematical Analysis and Applications)

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (4)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI