Class Of Degenerate Elliptic Equations Research Articles

A-priori bound and Hölder continuity of solutions to degenerate elliptic equations with variable exponents

We investigate the boundedness and regularity of solutions to degenerate elliptic equations with variable exponents that are subject to the Dirichlet boundary condition. By employing the De Giorgi iteration, we obtain a-priori bounds and the Hölder continuity for solutions. As an application, we obtain the existence of infinitely many small solutions for a class of degenerate elliptic equations involving variable exponents.

Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators

In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

Boundary Hölder Estimates for a Class of Degenerate Elliptic Equations in Piecewise Smooth Domains

Boundary Hölder Estimates for a Class of Degenerate Elliptic Equations in Piecewise Smooth Domains

The thin obstacle problem for some variable coefficient degenerate elliptic operators

In this paper, we establish the optimal interior regularity and the C1,γ smoothness of the regular part of the free boundary in the thin obstacle problem for a class of degenerate elliptic equations with variable coefficients.

Estimates of Eigenvalues of a Semiperiodic Dirichlet Problem for a Class of Degenerate Elliptic Equations

In this paper, we consider a class of degenerate elliptic equations with arbitrary power degeneration. The issues about the existence, uniqueness, and smoothness of solutions of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration are studied. The two-sided estimates for singular numbers (s-numbers) are obtained. Note that estimates of singular numbers (s-numbers) show the rate of approximation of the found solutions by finite-dimensional subspaces. Here, we also obtain estimates for the eigenvalues. We note that, in this paper, apparently, two-sided estimates of singular numbers (s-numbers) for degenerate elliptic operators are obtained for the first time. At the end of the paper, a symmetric operator is considered, i.e., a self-adjoint case.

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

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

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

Variational Integral and Some Inequalities of a Class of Quasilinear Elliptic System

This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincare inequalities for the case $$q<1$$ .

Existence results for nonlinear degenerate elliptic equations with lower order terms

Abstract In this paper, we prove the existence and regularity of solutions of the homogeneous Dirichlet initial-boundary value problem for a class of degenerate elliptic equations with lower order terms. The results we obtained here, extend some existing ones of [2, 9, 11] in some sense.

The Dirichlet problem for fully nonlinear degenerate elliptic equations with a singular nonlinearity

We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive viscosity solutions.

Multiple solutions for a class of infinitely degenerate elliptic equations with a sign‐changing weight function

In this paper, we study the Dirichlet problem for a class of degenerate elliptic equations with a sign‐changing weight function. Under some conditions on the weight function, we obtain at least two nontrivial solutions by the method of Nehari manifold and the logarithmic Sobolev inequality.

A Class of Degenerate Elliptic Equations With Nonlinear Boundary Conditions - WITHDRAWN

A Class of Degenerate Elliptic Equations With Nonlinear Boundary Conditions - WITHDRAWN

A class of degenerate elliptic equations with nonlinear boundary conditions – WITHDRAWAL

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Boundary blow-up solutions to a class of degenerate elliptic equations

Let \(\Omega \) be a bounded domain in \(\mathbb {R}^N=\mathbb {R}^{N_1} \times \mathbb {R}^{N_2}\) with \(N_1, N_2 \ge 1\), and \(N(s) = N_1 + (1+s)N_2\) be the homogeneous dimension of \(\mathbb {R}^N\) for \(s \ge 0\). In this paper, we prove the existence and uniqueness of boundary blow-up solutions to the following semilinear degenerate elliptic equation $$\begin{aligned} {\left\{ \begin{array}{ll} G_s u = {|x|^{2s}} u^p_+ \; &{}\text { in } \Omega ,\\ u(z)\rightarrow +\infty \; &{}\text { as } {d}(z) \rightarrow 0, \end{array}\right. } \end{aligned}$$ where \(u_+ = \max \{u,0\}\), \(1<p<{{N(s)+2s} \over {N(s)-2}}\), and d(z) denotes the Grushin distance from z to the boundary of \(\Omega \). Here \(G_s\) is the Grushin operator of the form $$\begin{aligned} G_s u= \Delta _x u + |x|^{2s}\Delta _y u, \; s\ge 0. \end{aligned}$$ It is worth noticing that our results do not require any assumption on the smoothness of the domain \(\Omega \), and when \(s=0\), we cover the previous results for the Laplace operator \(\Delta \).

Existence of solutions for a class of degenerate elliptic equations in $P(x)$-Sobolev spaces

We study the Dirichlet problem for degenerate elliptic equations of the form \begin{equation*} - \mbox{div} a(x,u,\nabla u)+ H(x,u,\nabla u)= f \quad \mbox{in } \Omega, \end{equation*} where $ a(x,u,\nabla u)$ is allowed to degenerate with respect to the unknown $u$, and $H(x,u,\nabla u)$ is a nonlinear term without sign condition. Under suitable conditions on $a$ and $H$, we prove the existence of bounded and unbounded solution for a datum $f\in L^m$, with $1\leq m\leq \infty$.

On a Class of Quasilinear Elliptic Equations with Degenerate Coerciveness and Measure Data

Abstract We study the existence of measure-valued solutions for a class of degenerate elliptic equations with measure data. The notion of solution is natural, since it is obtained by a regularization procedure which also relies on a standard approximation of the datum μ. We provide partial uniqueness results and qualitative properties of the constructed solutions concerning, in particular, the structure of their diffuse part with respect to the harmonic-capacity.

Regularity theorems for a class of degenerate elliptic equations

Regularity theorems for a class of degenerate elliptic equations

An excess-decay result for a class of degenerate elliptic equations

We consider a family of degenerate elliptic equations of the form div $(\nabla F(\nabla u)) = f$, where $F\in C^{1,1}$ is a convex function which is elliptic outside a ball.We prove an excess-decay estimate at points where $\nabla u$ is close to a nondegenerate value for $F$. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of $\nabla u$ outside the degeneracy, and to anisotropic versions of the $p$-laplacian, where we get Hölder regularity of $\nabla u$.

Hölder Estimates for A Class of Degenerate Elliptic Equations

In this paper we study the regularity theory of the solutions of a class of degenerate elliptic equations in divergence form. By introducing a proper distance and applying the compactness method we establish the Hölder type estimates for the weak solutions.

A priori estimates for a class of degenerate elliptic equations

In this paper we investigate the regularity of solutions for the following degenerate partial differential equation $$\left \{\begin{array}{ll} -\Delta_p u + u = f \qquad {\rm in} \,\Omega, \frac{\partial u}{\partial \nu} = 0 \qquad \qquad \,\,\,\,\,\,\,\,\,\, {\rm on} \,\partial \Omega, \end{array}\right.$$ when \({f \in L^q(\Omega), p > 2}\) and q ≥ 2. If u is a weak solution in \({W^{1, p}(\Omega)}\), we obtain estimates for u in the Nikolskii space \({\mathcal{N}^{1+2/r,r}(\Omega)}\), where r = q(p − 2) + 2, in terms of the Lq norm of f. In particular, due to imbedding theorems of Nikolskii spaces into Sobolev spaces, we conclude that \({\|u\|^r_{W^{1 + 2/r - \epsilon, r}(\Omega)} \leq C(\|f\|_{L^q(\Omega)}^q + \| f\|^{r}_{L^q(\Omega)} + \|f\|^{2r/p}_{L^q(\Omega)})}\) for every \({\epsilon > 0}\) sufficiently small. Moreover, we prove that the resolvent operator is continuous and compact in \({W^{1,r}(\Omega)}\).