Degenerate Elliptic Operators Research Articles

Critical points of solutions of sub-elliptic equations on the Heisenberg group

This paper concerns the critical points of a class of degenerate elliptic operators. We prove that the nonexistence of critical point of the Green function for the Kohn-Laplacian of an H-star-shaped region on the first Heisenberg group. Moreover, we prove that solutions to a class of semi-linear degenerate elliptic equations on an H-star-shaped ring have no critical point.

Estimates of Dirichlet eigenvalues for degenerate $$\triangle _{\mu }$$-Laplace operator

Let $$\varOmega $$ be a bounded open domain in $$\mathbb {R}^n$$ and $$\triangle _{\mu }=\sum _{j=1}^{n}\partial _{x_{j}}(\mu _{j}^2(x)\partial _{x_{j}})$$ be a class of degenerate elliptic operator with continuous nonnegative coefficients $$\mu _{1},\mu _{2},\ldots ,\mu _{n}$$ . Denote by $$\lambda _{k}$$ the kth Dirichlet eigenvalue of the self-adjoint degenerate elliptic operator $$-\triangle _{\mu }$$ on $$\varOmega $$ . If the coefficients $$\mu _{1},\mu _{2},\ldots ,\mu _{n}$$ satisfy some general assumptions, we give an explicit lower bound estimate of $$\lambda _{k}$$ . Moreover, if the coefficients $$\mu _{1}\ldots ,\mu _{n}$$ are homogeneous functions with respect to a group of dilation, then we obtain an explicit sharp lower bound estimate for $$\lambda _{k}$$ , which has a polynomially growth in k of the order related to the homogeneous dimension. Finally, we also establish an upper bound estimate of $$\lambda _{k}$$ for general self-adjoint degenerate elliptic operator $$\triangle _{\mu }$$ .

Regularity up to the boundary for some degenerate elliptic operators

ABSTRACTWe consider a class of severely degenerate elliptic operators and establish a Harnack inequality up to the boundary. As a consequence, we get smoothness of the weak solutions under very sharp assumptions regarding the lower order terms of the operator.

Regularity Properties for a Class of Non-uniformly Elliptic Isaacs Operators

Abstract We consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such an operator generally enjoys the qualitative properties of the Laplace operator, as for instance maximum and comparison principles, ABP and Harnack inequalities, Liouville theorems for subsolutions or supersolutions. Existence and uniqueness for the Dirichlet problem are also proved as well as local and global Hölder estimates for viscosity solutions. All results are discussed for a more general class of weighted partial trace operators.

Towards a reversed Faber–Krahn inequality for the truncated Laplacian

We consider the nonlinear eigenvalue problem, with Dirichlet boundary condition, for the very degenerate elliptic operator \mathcal{P}^+_{1} mapping a function u to the maximum eigenvalue of its Hessian matrix. The aim is to show that, at least for square type domains having fixed volume, the symmetry of the domain maximizes the principal eigenvalue, contrary to what happens for the Laplacian.

Vertical square functions and other operators associated with an elliptic operator

We study the vertical and conical square functions defined via elliptic operators in divergence form. In general, vertical and conical square functions are equivalent operators just in L2. But when this square functions are defined through the heat or Poisson semigroups that arise from an elliptic operator, we are able to find open intervals containing 2 where the equivalence holds. The intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off-diagonal estimates. As a consequence we obtain new boundedness results for some square functions. Besides, we consider a non-tangential maximal function associated with the Poisson semigroup and extend the known range where that operator is bounded. Our methods are based on the use of extrapolation for Muckenhoupt weights and change of angle estimates. All our results are obtained in the general setting of a degenerate elliptic operator, where the degeneracy is given by an A2 weight, in weighted Lebesgue spaces. Of course they are valid in the unweighted and/or non-degenerate situations, which can be seen as special cases, and they provide new results even in those particular settings.We also consider the square root of a degenerate elliptic operator in divergence form Lw and improve the lower bound of the interval where this operator is known to be bounded on Lp(vdw). Finally, we give unweighted boundedness results for the degenerate operators under consideration.

Global Existence, Exponential Decay and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Parabolic Equations

In this paper, we study the initial-boundary value problem for the semilinear parabolic equations ut − ∆Xu = |u|p−1u, where X = (X1, X2, ⋯ , Xm) is a system of real smooth vector fields which satisfy the Hormander’s condition, and $$\Delta_X\;=\;\sum\limits_{j = 1}^m\;{X_j^2}$$ finitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Finally, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy, and also we discuss the asymptotic behavior of the global solutions.

Spectral approximations of strongly degenerate elliptic differential operators

We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space $L_q(\Omega)$ on a bounded domain $\Omega$. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in $L_q(\Omega)$. The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional $E\left(t,u\right)$, which characterizes the shortest distance from an arbitrary function ${u\in L_q(\Omega)}$ to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed ${t>0}$. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than ${t>0}$. The approximation functional $E\left(t,u\right)$ in our cases plays a similar role as the modulus of smoothness in the functions theory.

Core properties for degenerate elliptic operators with complex bounded coefficients

Let ckl∈W2,∞(Rd,C) for all k,l∈{1,…,d}. We consider the divergence form operator A=−∑k,l=1d∂l(ckl∂k) in L2(Rd) when the coefficient matrix satisfies (C(x)ξ,ξ)∈Σθ for all x∈Rd and ξ∈Cd, where Σθ be the sector with vertex 0 and semi-angle θ in the complex plane. We show that for all p in a suitable interval the contraction semigroup generated by −A extends consistently to a contraction semigroup on Lp(Rd). For those values of p we present a condition on the coefficients such that the space Cc∞(Rd) of test functions is a core for the generator on Lp(Rd). We also examine the operator A separately in the more special Hilbert space L2(Rd) setting and provide more sufficient conditions such that Cc∞(Rd) is a core.

A New Elliptic Measure on Lower Dimensional Sets

The recent years have seen a beautiful breakthrough culminating in a comprehensive understanding of certain scale-invariant properties of n − 1 dimensional sets across analysis, geometric measure theory, and PDEs. The present paper surveys the first steps of a program recently launched by the authors and aimed at the new PDE approach to sets with lower dimensional boundaries. We define a suitable class of degenerate elliptic operators, explain our intuition, motivation, and goals, and present the first results regarding absolute continuity of the emerging elliptic measure with respect to the surface measure analogous to the classical theorems of C. Kenig and his collaborators in the case of co-dimension one.

On the strong unique continuation property of a degenerate elliptic operator with Hardy-type potential

In this paper, we prove the strong unique continuation property for the following degenerate elliptic equation 0.1$$\begin{aligned} \Delta _zu +|z|^2\partial _t^2u = Vu,\quad (z,t) \in {\mathbb {R}}^N \times {\mathbb {R}} \end{aligned}$$where the potential V satisfies either of the following growth assumptions 0.2$$\begin{aligned}&\left| V(z,t) \right| \le \frac{f(\rho (z,t))}{\rho (z,t)^2}, \text {where} \rho \text { is as in }(2.1)\text { and }\nonumber \\&\quad f\text { satisfies the Dini integrability condition as in }(1.3). \end{aligned}$$or when $$\begin{aligned}&\left| V(z,t) \right| \le C \frac{\psi (z,t)^{\epsilon }}{\rho (z,t)^2}, \text {for some }\epsilon >0\text { with }\psi \text { as in }(2.6)\text { and } N\text { even.} \end{aligned}$$This extends some of the previous results obtained in [18] for this subfamily of Baouendi–Grushin operators. As corollaries, we obtain new unique continuation properties for solutions u to $$\begin{aligned} \Delta _{{\mathbb {H}}} u = Vu \end{aligned}$$with certain symmetries as expressed in (1.6) where $$\Delta _{{\mathbb {H}}}$$ corresponds to the sub-Laplacian on the Heisenberg group $${\mathbb {H}}^n$$.

Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

In this paper, we study the initial-boundary value problem for the semilinear pseudo-parabolic equations ut − ΔXut − ΔXu = |u|p−1u, where X = (X1,X2,…,Xm) is a system of real smooth vector fields which satisfy the Hormander’s condition, and ΔX = $${{\rm{\Delta}}_X} = \sum\nolimits_{j = 1}^m {X_j^2} $$ is a finitely degenerate elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.

Min–Max formulas for nonlocal elliptic operators on Euclidean Space

An operator satisfies the Global Comparison Property if anytime a function touches another from above at some point, then the operator preserves the ordering at the point of contact. This is characteristic of degenerate elliptic operators, including nonlocal and nonlinear ones. In previous work, the authors considered such operators in Riemannian manifolds and proved they can be represented by a min–max formula in terms of Lévy operators. In this note we revisit this theory in the context of Euclidean space. With the intricacies of the general Riemannian setting gone, the ideas behind the original proof of the min–max representation become clearer. Moreover, we prove new results regarding operators that commute with translations or which otherwise enjoy some spatial regularity.

Dahlberg's theorem in higher co-dimension

In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n−1 in Rn, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Γ of dimension d in Rn, d<n−1, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure ωL is absolutely continuous with respect to the Hausdorff measure on Γ. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of ωL and the Hausdorff measure.

Construction of Exact Solutions of Irregularly Degenerate Elliptic Equations with Analytic Coefficients

We solve a boundary value problem (problem E in the sense of M.V. Keldysh) for an irregularly degenerate elliptic operator in a rectangle. The exact solution of the problem is constructed as a series in the eigenfunctions of the limit operator. The method of spectral isolation of singularities, which generalizes the method of regularization of singular perturbations to the case of degenerate elliptic equations, is developed.

W2,p a priori estimates for nonvariational operators: the sharp maximal function technique

We consider a nonvariational degenerate elliptic operator, structured on a system of left invariant, 1-homogeneous, Hormander vector fields on a Carnot group, where the coefficient matrix is symmetric, uniformly positive on a bounded domain and the coefficients are locally VMO. We discuss a new proof (given in [BT] and also based on results in [BF]) of the interior estimates in Sobolev spaces, first proved in [BB-To]. The present proof extends to this context Krylov' technique, introduced in [K1], consisting in estimating the sharp maximal function of second order derivatives.

On $$A_p$$ A p weights and the Landau equation

In this manuscript we investigate the regularization of solutions for the spatially homogeneous Landau equation. For moderately soft potentials, it is shown that weak solutions become smooth instantaneously and stay so over all times, and the estimates depend only on the initial mass, energy, and entropy. For very soft potentials we obtain a conditional regularity result, hinging on what may be described as a nonlinear Morrey space bound, assumed to hold uniformly over time. This bound always holds in the case of very soft potentials, and nearly holds for general potentials, including Coulomb. This latter phenomenon captures the intuition that for very soft potentials, the dissipative term in the equation is of the same order as the quadratic term driving the growth (and potentially, singularities). In particular, for the Coulomb case, the conditional regularity result shows a rate of regularization much stronger than what is usually expected for regular parabolic equations. The main feature of our proofs is the analysis of the linearized Landau operator around an arbitrary and possibly irregular distribution. This linear operator is shown to be a degenerate elliptic Schrodinger operator whose coefficients are controlled by $$A_p$$ -weights.

Asymptotic behavior and blow-up of solutions for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity

In this paper, we study the initial-boundary value problem for infinitely degenerate semilinear parabolic equations with logarithmic nonlinearity ut−△Xu=ulog⁡|u|, where X=(X1,X2,⋯,Xm) is an infinitely degenerate system of vector fields, and △X:=∑j=1mXj2 is an infinitely degenerate elliptic operator. Using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the logarithmic Sobolev inequality, we obtain the global existence and blow-up at +∞ of solutions with low initial energy or critical initial energy, and we also discuss the asymptotic behavior of the solutions.

On Some Degenerate Elliptic Equations Arising in Geometric Problems

We consider some fully nonlinear degenerate elliptic operators and we investigate the validity of certain properties related to the maximum principle. In particular, we establish the equivalence between the sign propagation property and the strict positivity of a suitably defined generalized principal eigenvalue. Furthermore, we show that even in the degenerate case considered in the present paper, the well-known condition introduced by Keller–Osserman on the zero-order term is necessary and sufficient for the existence of entire weak subsolutions.

Strong comparison principles for some nonlinear degenerate elliptic equations

In this paper, we obtain the strong comparison principle and Hopf Lemma for locally Lipschitz viscosity solutions to a class of nonlinear degenerate elliptic operators of the form ∇2 ψ + L(x, ∇ ψ), including the conformal hessian operator.