Solutions Of Degenerate Elliptic Equations Research Articles

Existence and regularity of solutions for degenerate elliptic equations with variable growth

Existence and regularity of solutions for degenerate elliptic equations with variable growth

Existence and approximation of solutions for a class of degenerate elliptic equations with Neumann boundary condition

In this work we study the equation , where is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set . We prove the existence and uniqueness of weak solutions in the weighted Sobolev space for the Neumann problem. The main result establishes that a weak solution of degenerate elliptic equations can be approximated by a sequence of solutions for non-degenerate elliptic equations

Weighted [formula omitted] estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable

This paper studies Sobolev regularity of weak solution of degenerate elliptic equations in divergence form div[A(X)∇u]=div[F(X)], where X=(x,y)∈Rn×R. The coefficient matrix A(X) is a symmetric, measurable (n+1)×(n+1) matrix, and it could be degenerate or singular in the one dimensional y-variable as a weight function in the A2 Muckenhoupt class. Our results give weighted Sobolev regularity estimates of Calderón–Zygmund type for weak solutions of this class of degenerate/singular equations. As an application of these estimates, we establish global fractional Sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients. This result can be considered as the Sobolev counterpart of the recently established Schauder regularity theory of fractional elliptic equations.

Multiple positive solutions for degenerate elliptic equations with singularity and critical cone Sobolev exponents

In this paper, we study the existence of multiple positive solutions for degenerate elliptic systems with singular and critical cone Sobolev exponents on singular manifolds. With the help of the variational method, we obtain a multiplicity result.

Liouville's theorem and comparison results for solutions of degenerate elliptic equations in exterior domains

A version of Liouville's theorem is proved for solutions of some degenerate elliptic equations defined in Rn\\K, where K is a compact set, provided the structure of this equation and the dimension n are related. Theoretical and numerical examples are given. Furthermore, a comparison result and the uniqueness of solution are obtained for such equations in exterior domains.

Homogenization of the Brush Problem with a Source Term in L 1

We consider a domain which has the form of a brush in 3D or the form of a comb in 2D, i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one teeth to another one and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to e, and the asymptotic volume fraction of the teeth (as e tends to zero) is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth. In this domain we study the asymptotic behavior (as e tends to zero) of the solution of a second order uniformly elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is satisfied on the whole of the boundary. First, we revisit the problem where the source term belongs to L^2. This is a classical problem, but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new. Then, we study the case where the source term belongs to L^1. Working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result.

Local Gradient Estimates for Degenerate Elliptic Equations

Abstract This paper is focused on the local interior W 1 , ∞ ${W^{1,\infty}}$ -regularity for weak solutions of degenerate elliptic equations of the form div ⁡ [ 𝐚 ⁢ ( x , u , ∇ ⁡ u ) ] + b ⁢ ( x , u , ∇ ⁡ u ) = 0 ${\operatorname{div}[\mathbf{a}(x,u,\nabla u)]+b(x,u,\nabla u)=0}$ , which include those of p-Laplacian type. We derive an explicit estimate of the local L ∞ ${L^{\infty}}$ -norm for the solution’s gradient in terms of its local L p ${L^{p}}$ -norm. Specifically, we prove ∥ ∇ ⁡ u ∥ L ∞ ⁢ ( B R / 2 ⁢ ( x 0 ) ) p ≤ C | B R ⁢ ( x 0 ) | ⁢ ∫ B R ⁢ ( x 0 ) | ∇ ⁡ u ⁢ ( x ) | p ⁢ 𝑑 x . $\lVert\nabla u\rVert_{L^{\infty}(B_{R/2}(x_{0}))}^{p}\leq\frac{C}{\lvert B_{R}% (x_{0})\rvert}\int_{B_{R}(x_{0})}\lvert\nabla u(x)\rvert^{p}dx.$ This estimate paves the way for our work [9] in establishing W 1 , q ${W^{1,q}}$ -estimates (for q > p ${q>p}$ ) for weak solutions to a much larger class of quasilinear elliptic equations.

Degenerate nonlinear elliptic equations lacking in compactness

In this paper, the authors prove the existence of solutions for degenerate elliptic equations of the form −div(a(x)∇pu(x)) = g(λ, x, |u|p−2u) in ℝN, where ∇pu = |∇u|p−2∇u and a(x) is a degenerate nonnegative weight. The authors also investigate a related nonlinear eigenvalue problem obtaining an existence result which contains information about the location and multiplicity of eigensolutions. The proofs of the main results are obtained by using the critical point theory in Sobolev weighted spaces combined with a Caffarelli-Kohn-Nirenberg-type inequality and by using a specific minimax method, but without making use of the Palais-Smale condition.

On a class of weighted Gauss-type isoperimetric inequalities and applications to symmetrization

We solve a class of weighted isoperimetric problems of the form $$ \mathrm {min}\left{\int\_{\partial E}w e^V,dx:\int\_E e^V,dx=\mathrm {constant}\right} $$ where $w$ and $V$ are suitable functions on $\mathbb R^d$. As a consequence, we prove a comparison result for the solutions of degenerate elliptic equations.

Hölder continuity of degenerate p-harmonic functions

We prove a partial Holder continuity result for finite energy solutions of degenerate elliptic equations. The function that measures the degeneracy of the problem is assumed to belong to a suitable Sobolev class. Moreover, we prove an analogous result for infinite energy solutions provided their gradients have a suitable degree of integrability.

Existence and some properties of solutions for degenerate elliptic equations with exponent variable

In this paper, we study degenerate elliptic equations with variable exponents when a perturbation term satisfies the Ambrosetti–Rabinowitz condition and does not satisfy the Ambrosetti–Rabinowitz condition. For the first case, we employ the standard Mountain Pass theorem to give the existence of solutions. For the second case, we use Browder’s theorem for monotone operators to show the unique existence of a solution when the perturbation term is decreasing with respect to a function variable. A priori bound and nonnegativeness of solutions are also given. We emphasize that the log-Hölder continuous condition is not required.

Existence of Infinite Energy Solutions of Degenerate Elliptic Equations

Existence of Infinite Energy Solutions of Degenerate Elliptic Equations

Existence of solutions for degenerate elliptic equations with singular potential on conical singular manifolds

AbstractIn this work, we study the Dirichlet problem for a class of degenerate elliptic equations with singular potential on conical singular manifolds. By using the cone Sobolev inequality and Hardy inequality, the existence of nontrivial solutions has been proved.

On the continuity of minimizers for quasilinear functionals

In this paper we establish a continuity result for local minimizers of some quasilinear functionals that satisfy degenerate elliptic bounds. The non-negative function which measures the degree of degeneracy is assumed to be exponentially integrable. The minimizers are shown to have a modulus of continuity controlled by log log(1/|x|)−1. Our proof adapts ideas developed for solutions of degenerate elliptic equations by J. Onninen, X. Zhong: Continuity of solutions of linear, degenerate elliptic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), 103–116.

Unique continuation for positive solutions of degenerate elliptic equations

We establish the strong unique continuation property for positive weak solutions to degenerate quasilinear elliptic equations. The degeneracy is given by a suitable power of a strong A∞ weight (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Critical points of solutions of degenerate elliptic equations in the plane

We study the minimizer u of a convex functional in the plane which is not Gâteaux-differentiable. Namely, we show that the set of critical points of any C 1-smooth minimizer can not have isolated points. Also, by means of some appropriate approximating scheme and viscosity solutions, we determine an Euler–Lagrange equation that u must satisfy. By applying the same approximating scheme, we can pair u with a function v which may be regarded as the stream function of u in a suitable generalized sense.

Maximum principles for weak solutions of degenerate elliptic equations with a uniformly elliptic direction

For second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L 2 -based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.

Regularity of the solutions of degenerate elliptic equations in divergent form

Ap rioriestimates of the solution to the Dirichlet problem and of its first derivatives in terms of weighted Lebesgue norms are obtained for linear and quasilinear equations with degeneracy from Ap Muckenhoupt classes. DOI: 10.1134/S000143460801001X

Homogenization of degenerate elliptic equations

We consider the divergent elliptic equations whose weight function and its inverse are assumed locally integrable. The equations of this type exhibit the Lavrentiev phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. We classify the weak solutions of degenerate elliptic equations and show the attainability of the so-called W-solutions. Investigating the homogenization of arbitrary attainable solutions, we find their different asymptotic behavior. Under the assumption of the higher integrability of the weight function we estimate the difference between the exact solution and certain special approximations.

Regularity results for degenerate elliptic systems

We prove regularity results for certain degenerate quasilinear elliptic systems with coefficients which depend on two different weights. By using Sobolev- and Poincaré inequalities due to Chanillo and Wheeden [S. Chanillo, R.L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, Amer. J. Math. 107 (1985) 1191–1226; S. Chanillo, R.L. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986) 1111–1134] we derive a new weak Harnack inequality and adapt an idea due to L. Caffarelli [L.A. Caffarelli, Regularity theorems for weak solutions of some nonlinear systems, Comm. Pure Appl. Math. 35 (1982) 833–838] to prove a priori estimates for bounded weak solutions. For example we show that every bounded weak solution of the system −Dα(aαβ(x,u,∇u)Dβui)=0 with |x|2|ξ|2⩽aαβξαξβ⩽|x|τ|ξ|2, |x|<1, τ∈(1,2) is Hölder continuous. Furthermore we derive a Liouville theorem for entire solutions of the above systems.