Regularity of weak solutions to certain degenerate elliptic equations (original) (raw)

An Lp-Estimate for Weak Solutions of Elliptic Equations

2012

and Applied Analysis 3 2. A Generalization of a Result by Stampacchia Let G : t ∈ R −→ G t 2.1 be a uniformly Lipschitz real function, such that there exists a positive constant K such that for every t′, t′′ ∈ R one has ∣G ( t′ ) −Gt′′∣ ≤ K∣t′ − t′′∣, 2.2 and suppose that G| −k,k 0, for a k ∈ R 2.3 and that its derivative G′ has a finite number of discontinuity points. A known result by Stampacchia, see Lemma1.1 in 1 or in 13 , for details , guarantees that given a function u, defined in an open bounded subset of R and belonging to ◦ W1,2, also the composition between G and u is in ◦ W1,2 and gives an explicit expression for the derivative of this composition, up to sets of null Lebesgue measure. Later on, in 7 , Bottaro and Marina explicitly observed that, up to few modifications, the proof of these results remains valid also for an unbounded open subset Ω of R, n ≥ 2. More precisely, u ∈ ◦ W1,2 Ω ⇒ G u G ◦ u ∈ ◦ W1,2 Ω , 2.4 and moreover G u xi G ′ u uxi, a. e. in Ω, i 1, . . . , ...

Existence and uniqueness of weak solution in weighted Sobolev spaces for a class of nonlinear degenerate elliptic problems with measure data

2021

In this paper, we study the existence and uniqueness of weak solution to a Dirichlet boundary value problems for the following nonlinear degenerate elliptic problems −div [ ω1A(x,∇u) + ν2B(x, u,∇u) ] + ν1C(x, u) + ω2|u|u = f − divF, where 1 < p < ∞, ω1, ν2, ν1 and ω2 are Ap-weight functions, and A : Ω × R −→ R, B : Ω×R×R −→ R, C : Ω×R −→ R are Caratéodory functions that satisfy some conditions and the right-hand side term f − divF belongs to Lp(Ω, ω ′ 2 ) + n ∏ j=1 L ′ (Ω, ω ′ 1 ). We will use the BrowderMinty Theorem and the weighted Sobolev spaces theory to prove the existence and uniqueness of weak solution in the weighted Sobolev space W 1,p 0 (Ω, ω1, ω2).

On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary

Discrete & Continuous Dynamical Systems - A, 2010

Notations and preliminary results. We shall always consider Ω ⊂ IR N , N 2, a bounded open set of class C 2,1. For any measurable set E ⊂ IR N we shall denote by |E| its Lebesgue measure. We shall consider a linear operator L : Lu = − N i,j=1 ∂ i (a ij (x)∂ j u). We assume that a ij = a ji ∈ C 0,1 (Ω), ∀ξ = (ξ 1 ,. .. , ξ N) ∈ IR N , and that L is elliptic in the sense that i,j a ij (x)ξ i ξ j α 0 |ξ| 2 for some α 0 > 0.

A Kolmogorov-Szego-Krein type condition for weighted Sobolev spaces

Indiana University Mathematics Journal, 2005

§0. Introduction and main results Let p ∈ [1, +∞), k ∈ N, and let µ = (µ 0 , µ 1 ,. .. , µ k) be a (k + 1)-tuple of positive finite Borel measures on the unit circle T = {z : |z| = 1} in the complex plane. Consider the continuous mapping Π : C k (T) → k j=0 C(T), given by Πf = f, f ,. .. , f (k) , where f (z) = df dz (all spaces of functions that we consider are complex-valued). Note that df dz (e iθ) = −ie −iθ d dθ f (e iθ). Definition. The abstract Sobolev space W k,p (µ) = W k,p (µ 0 ,. .. , µ k) is the closure of ΠC k (T) in the space k j=0 L p (T, µ j). We refer to [17] for the classical theory of Sobolev spaces in domains of R n. We refer to [13, 18, 5, 12] for the theory of weighted Sobolev spaces in domains of R n ; in [11, 14], one can find applications of this topic to partial differential equations. We consider in W k,p (µ) the usual norm f k,p,µ = k j=0 f j p p,µ j 1/p , f = (f 0 ,. .. , f k). Each function f in C k (T) has its image Πf in W k,p (µ), and these images are dense in W k,p (µ). In many cases, an element g = (g 0 ,. .. , g k) in W k,p (µ) is completely determined by its first component g 0 , so that W k,p (µ) can be identified with a certain space of measurable functions g 0 , and the components g 1 ,. .. , g k can be thought of as a kind of generalized derivatives of g 0. In general, however, elements of W k,p (µ) cannot be identified with scalar functions on T. This setting of abstract Sobolev spaces is the most natural for us. (See [2], [22]-[26] in order to know when W k,p (µ) is a space of functions.) This space plays a central role in the theory of orthogonal polynomials with respect to Sobolev inner products (see [2], [15], [16] and [23]; in [2] and [16], the authors consider measures supported in compact sets in the complex plane). In fact, if the multiplication operator (M f)(z) = zf (z) is bounded in W k,2 (µ), then every 1 The research of the first author was partially supported by two grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. 2 The research of the second author was partially supported by the Ramón y Cajal Programme by the Ministry of Science and Technology of Spain.

Some Remarks About the Density of Smooth Functions in Weighted Sobolev Spaces

1994

Introduction and statement of main resultsIn this note we deal with the problem of the density of smooth functions in weightedSobolev spaces (for general results and references on this topic see, for instance, [14], [10],[2], and the bibliography therein). In order to introduce some definitions, let us fix abounded openset\Omega` IRn, a real number p ? 1, and a function

P A ON SUB WEAKLY ω-CONTINUOUS FUNCTIONS

On sub weakly w continuous functions, 2018

The purpose of this paper is to introduce a new class functions called, sub weakly ω-continuous functions. Also, we obtain its characterizations and its basic properties.

Existence and regularity theory in weighted Sobolev spaces and applications

2016

In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincaré inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form: { div (ρ(x)|∇u|p−2∇u) = x∗, u− w ∈ W 1,p ρ,0 (Ω), (0.0.1) involving any given x∗ ∈ (W 1,p ρ,0 (Ω))∗ and w ∈ W 1,p ρ (Ω), where u ∈ W 1,p ρ (Ω) and W 1,p ρ (Ω) denotes certain weighted Sobolev space, W 1,p ρ,0 (Ω) is the completion of C∞ 0 (Ω). As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDE...

On a weighted variable spaces $ L_ {p (x),\ omega} $ for 0<p(x)<10< p (x)< 10<p(x)<1 and weighted Hardy inequality

On a weighted variable spaces L p(x), ω for 0 < p(x) < 1 and weighted Hardy inequality ROVSHAN A.BANDALIEV ABSTRACT. In this paper a weighted variable exponent Lebesgue spaces L p(x), ω for 0 < p(x) < 1 is investigated. We show that this spaces is a quasi-Banach spaces. Note that embedding theorem between weight variable Lebesgue spaces is proved. In particular, we show that L p(x), ω (Ω) for 0 < p(x) < 1 isn't locally convex. Also, in this paper a some two-weight estimates for Hardy operator are proved.