Degenerate Elliptic Operators Research Articles

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth

Reiterated Homogenization of nonlinear degenerate elliptic operators with nonstandard growth

Microlocal analysis theory of degenerate elliptic operators

Microlocal analysis theory of degenerate elliptic operators

Boundary triples for a family of degenerate elliptic operators of Keldysh type

Boundary triples for a family of degenerate elliptic operators of Keldysh type

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Dirichlet problem for a class of nonlinear degenerate elliptic operators with critical growth and logarithmic perturbation

Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form L1(x)+g(x)L2(y), where L1(x) is one-dimensional and g(x) may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation, and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our methods do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.

Comparison principles for nonlinear potential theories and PDEs with fiberegularity and sufficient monotonicity

We present some recent advances in the productive and symbiotic interplay between general potential theories (subharmonic functions associated to closed subsets F⊂J2(X) of the 2-jets on X⊂Rn open) and subsolutions of degenerate elliptic and parabolic PDEs of the form F(x,u,Du,D2u)=0. We will implement the monotonicity-duality method begun by Harvey and Lawson (2009) (in the pure second order constant coefficient case) for proving comparison principles for potential theories where F has sufficient monotonicity and fiberegularity (in variable coefficient settings) and which carry over to all differential operators F which are compatible with F in a precise sense for which the correspondence principle holds.We will consider both elliptic and parabolic versions of the comparison principle in which the effect of boundary data is seen on the entire boundary or merely on a proper subset of the boundary.Particular attention will be given to gradient dependent examples with the requisite sufficient monotonicity of proper ellipticity and directionality in the gradient. Examples operators we will discuss include the degenerate elliptic operators of optimal transport in which the target density is strictly increasing in some directions as well as operators which are weakly parabolic in the sense of Krylov. Further examples, modeled on hyperbolic polynomials in the sense of Gårding give a rich class of examples with directionality in the gradient. Moreover we present a model example in which the comparison principle holds, but standard viscosity structural conditions fail to hold.

Nontrivial solutions for an asymptotically linear Delta alpha-Laplace equation

In this paper, we study a class of degenerate unperturbed problems. We first investigate some properties of eigenvalues and eigenfunctions for the strongly degenerate elliptic operator and then obtain two existence theorems of nontrivial solutions when the nonlinearity is a function with an asymptotically condition.

Local Hopf Lemma for Certain Degenerate Elliptic Operators on the Half-Space

Local Hopf Lemma for Certain Degenerate Elliptic Operators on the Half-Space

Global regularity for a class of fully nonlinear PDEs with unbalanced variable degeneracy

AbstractWe establish the existence and sharp global regularity results (, and estimates) for a class of fully nonlinear elliptic Partial Differential Equations (PDEs) with unbalanced variable degeneracy. In a precise way, the degeneracy law of the model switches between two different kinds of degenerate elliptic operators of variable order, according to the null set of a modulating function . The model case in question is given by for a bounded, regular, and open set , and appropriate continuous data , , and . Such sharp regularity estimates generalize and improve, to some extent, earlier ones via geometric treatments. Our results are consequences of geometric tangential methods and make use of compactness, localized oscillating, and scaling techniques. In the end, our findings are applied in the study of a wide class of nonlinear models.

Measure concentration and the Schrödinger equation

AbstractThis talk pursued the aim to represent the solutions of the electronic Schrödinger equation as traces of higher‐dimensional functions. This allows to decouple the electron‐electron interaction potential but comes at the price of a degenerate elliptic operator replacing the Laplace operator on the higher‐dimensional space. The surprising observation is that this operator can without much loss again be substituted by the Laplace operator, the more successful the larger the system under consideration is. This is due to a nontrivial concentration of measure phenomenon that has much to do with the random projection theorem known from probability theory and can, for example, serve as a building block for the construction of iterative methods that map sums of products of orbitals and geminals onto functions of the same type.

Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains

Let n≥2, w be a Muckenhoupt A2(Rn) weight, Ω a bounded Lipschitz domain of Rn, and L:=−w−1div(A∇⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted Lp estimate for the Kato square root of L:‖L1/2(f)‖Lp(Ω,vw)∼‖∇f‖Lp(Ω,vw) for any f∈W01,p(Ω,vw) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,p(Ω,vw) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w, W01,p(Ω,vw) and W1,p(Ω,vw) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w, via letting v:=w−1, the unweighted L2 estimate for the Kato square root of L that ‖L1/2(f)‖L2(Ω)∼‖∇f‖L2(Ω) for any f∈W01,2(Ω) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,2(Ω) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L2 estimates, the unweighted L2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established.

Fully nonlinear Hamilton–Jacobi equations of degenerate type

We examine Hamilton–Jacobi equations driven by fully nonlinear degenerate elliptic operators in the presence of superlinear Hamiltonians. By exploring the Ishii–Jensen inequality, we prove that viscosity solutions are locally Lipschitz-continuous, with estimates depending on the structural conditions of the problem. We close the paper with an application of our findings to a two-phase free boundary problem.

Approximation of Green functions and domains with uniformly rectifiable boundaries of all dimensions

The present paper concerns divergence form elliptic and degenerate elliptic operators in a domain Ω⊂Rn, and establishes the equivalence between the uniform rectifiability of the boundary E=∂Ω and weak Carleson condition on the good approximation of the Green function G by affine, or distance, functions. There are two main original contexts for the results, elliptic operators in a non-tangential access domain with an n−1 dimensional boundary and degenerate elliptic operators adapted to a domain with an Ahlfors regular boundary of larger co-dimension. In both cases necessary and sufficient conditions are given, in the form of Carleson packing conditions on the collection of balls centered on E where G is not well approximated.(1)This is the first time the underlying property of the control of the Green function by affine functions, or by the distance to the boundary, in the sense of the Carleson prevalent sets, appears in the literature; some results established here are new even in the half space;(2)the results are optimal, providing a full characterization of uniform rectifiability under the (standard) mild topological assumptions;(3)to the best of the authors' knowledge, even in traditional domains with (n−1)-dimensional boundaries, this is the first free boundary result applying to all elliptic operators, without any restriction on the coefficients (the direct one assumes the standard, and necessary, Carleson measure condition);(4)this is the first free boundary result in higher co-dimensional setting and as such, the first PDE characterization of uniform rectifiability for a set of dimension d, d<n−1, in Rn. The paper offers a general way to deal with related issues considerably beyond the scope of the aforementioned theorem, including the question of approximability of the gradient of the Green function, and the comparison of the Green function to a certain version of the distance to the original set rather than distance to the hyperplanes.

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.

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

This paper is concerned with the existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, the precise lower bound estimate of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.

Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

In the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If Omega is an open domain in mathbb {R}^n with mild topological conditions, we can even characterize the n-1 dimensional uniformly rectifiability of the boundary partial Omega by the A_infty -absolute continuity of the harmonic measure on partial Omega with respect to the surface measure. In low dimension (d<n-1), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if Gamma is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on Gamma is A_infty -absolutely continuous with respect to the d-dimensional Hausdorff measure. In the present article, we use a completely new approach to give an alternative and significantly shorter proof of David and Mayboroda’s result.

Existence of ground state solution for semilinear -Laplace equation

ABSTRACT This paper explores the existence of ground state solutions for a class of semilinear -Laplace equations where is a coercive potential, is a strongly degenerate elliptic operator, f is a function with more general superlinear conditions or asymptomatic linear conditions.

ВАРИАЦИОННАЯ ЗАДАЧА ДИРИХЛЕ С НЕОДНОРОДНЫМИ ГРАНИЧНЫМИ УСЛОВИЯМИ ДЛЯ ВЫРОЖДАЮЩИХСЯ ЭЛЛИПТИЧЕСКИХ ОПЕРАТОРОВ

We study the variational Dirichlet problem with nonhomogeneous boundary conditions for degenerate elliptic operators in a bounded domain with summable lower coefficients in the case when the corresponding sesquilinear forms may not satisfy the coercivity condition. A case is singled out when nonhomogeneous boundary conditions are specified explicitly and their number depends on the degree of degeneration of the leading coefficients of the operator under study. An inequality is proved in which the norm of the solution of the nonhomogeneous variational Dirichlet problem is estimated from above by the norms of the boundary functions and the right-hand side of the equation.

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.

The regularity problem for degenerate elliptic operators in weighted spaces

We study the solvability of the regularity problem for degenerate elliptic operators in the block case for data in weighted spaces. More precisely, let L_w be a degenerate elliptic operator with degeneracy given by a fixed weight w\in A_2(dx) in \mathbb{R}^n , and consider the associated block second order degenerate elliptic problem in the upper-half space \mathbb{R}_+^{n+1} . We obtain non-tangential bounds for the full gradient of the solution of the block case operator given by the Poisson semigroup in terms of the gradient of the boundary data. All this is done in the spaces L^p(vdw) , where v is a Muckenhoupt weight with respect to the underlying natural weighted space (\mathbb{R}^n, wdx) . We recover earlier results in the non-degenerate case (when w\equiv 1 , and with or without weight v ). Our strategy is also different and more direct thanks in particular to recent observations on change of angles in weighted square function estimates and non-tangential maximal functions. Our method gives as a consequence the (unweighted) L^2(dx) -solvability of the regularity problem for the block operator \mathbb{L}_\alpha u(x,t) = -|x|^{\alpha} \mathrm{\operatorname{div}}_x \big(|x|^{-\alpha } \,A(x) \nabla_x u(x,t)\big)-\partial_{t}^2 u(x,t) for any complex-valued uniformly elliptic matrix A and for all -\varepsilon&lt;\alpha&lt;{2n}/(n+2) , where \varepsilon depends just on the dimension and the ellipticity constants of A .