Degenerate Elliptic Operators Research Articles

Entropy and approximation numbers of embeddings of weighted Sobolev spaces

This paper examines the asymptotic behaviour of entropy and approximation numbers of compact embeddings of weighted Sobolev spaces into Lebesgue spaces. We consider admissible weights multiplied by a polynomial. The main results are related to the distribution of eigenvalues of some degenerate elliptic operators.

A Priori Boundary Estimations for an Elliptic Operator

AbstractIn this article, we consider a singular and a degenerate elliptic operators in a divergence form. The singularities exist on a part ofboundary, and comparable to the logarithmic distance function or its inverse. If we assume that the operator can be treated in apointwise sense than distribution sense, with this operator we obtain a priori Harnack continuity near the boundary. In the proof we transform the singular elliptic operator to uniformly bounded elliptic operator with unbounded first order terms. We study this type ofestimations considering a De Giorgi conjecture. In his conjecture, he proposed a certain ellipticity condition to guarantee a c ontinuity of a solution. Key words : Partial Differential Equations, Singular Elliptic, Degenerate Elliptic, De Giorgi Conjecture 1. Introduction 1 We consider the second order, linear, elliptic partial differential equation with divergence structure:(D)


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Maximum Principle and generalized principal eigenvalue for degenerate elliptic operators

We characterize the validity of the Maximum Principle in bounded domains for fully nonlinear degenerate elliptic operators in terms of the sign of a suitably defined generalized principal eigenvalue. Here, the maximum principle refers to the property of non-positivity of viscosity subsolutions of the Dirichlet problem. The new notion of generalized principal eigenvalue that we introduce here allows us to deal with arbitrary type of degeneracy of the elliptic operators. We further discuss the relations between this notion and other natural generalizations of the classical notion of principal eigenvalue, some of which have been previously introduced for particular classes of operators.

On the sectoriality of a class of degenerate elliptic operators arising in population genetics

We study the sectoriality of a class of degenerate second-order elliptic differential operators in the space of continuous functions on the canonical simplex of \({\mathbb{R}^{d}}\) . This kind of operators arises from the theory of Fleming–Viot processes in population genetics.

Bifurcation at isolated singular points for a degenerate elliptic eigenvalue problem

We consider bifurcation from the line of trivial solutions for a nonlinear eigenvalue problem on a bounded open subset, Ω, of RN with N≥3, containing 0. The leading term is a degenerate elliptic operator of the form L(u)=∇⋅A∇u where A∈C(Ω¯) with A>0 on Ω¯∖{0} and limx→0A(x)|x|2∈(0,∞). Solutions should satisfy u=0 on ∂Ω and the energy associated with L should be finite: ∫ΩA|∇u|2dx<∞. The nonlinear terms are of lower order, depending only on u and ∇u. Under our hypotheses the associated Nemytskii operators are not Fréchet differentiable at the trivial solution u=0.

An Extension of Hörmander’s Hypoellipticity Theorem

Motivated by applications to stochastic differential equations, an extension of Hormander’s hypoellipticity theorem is proved for second-order degenerate elliptic operators with non-smooth coefficients. The main results are established using point-wise Bessel kernel estimates and a weighted Sobolev inequality of Stein and Weiss. Of particular interest is that our results apply to operators with quite general first-order terms.

Lower and upper bounds of Dirichlet eigenvalues for totally characteristic degenerate elliptic operators

Let λ k be the k-th Dirichlet eigenvalue of totally characteristic degenerate elliptic operator $ - \Delta _\mathbb{B} $ defined on a stretched cone $\mathbb{B}_0 \subseteq [0,1) \times X$ with boundary on {x 1 = 0}. More precisely, $\Delta _\mathbb{B} = (x_1 \partial _{x_1 } )^2 \partial _{x_2 }^2 + \cdots + \partial _{x_n }^2 $ is also called the cone Laplacian. In this paper, by using Mellin-Fourier transform, we prove that $\lambda _k \geqslant C_n k^{\tfrac{2} {n}} $ for any k ⩾ 1, where $C_n = (\tfrac{n} {{n + 2}})(2\pi )^2 (|\mathbb{B}_0 |B_n )^{ - \tfrac{2} {n}} $ , which gives the lower bounds of the Dirchlet eigenvalues of $ - \Delta _\mathbb{B} $ . On the other hand, by using the Rayleigh-Ritz inequality, we deduce the upper bounds of λ k , i.e., $\lambda _{k + 1} \leqslant (1 + \tfrac{4} {n})k^{2/n} \lambda _1 $ . Combining the lower and upper bounds of λ k , we can easily obtain the lower bound for the first Dirichlet eigenvalue $\lambda _1 \geqslant C_n (1 + \tfrac{4} {n})^{ - 1} 2^{\tfrac{n} {2}} $ .

Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth

We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth. One fundamental in this topic is to extend the classical compactness results of the Σ-convergence method to the Orlicz spaces. We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.

A degenerate elliptic operator with unbounded diffusion coefficients

We prove that, for N \ge 3 , the operator L=|x|^\alpha\Delta generates an analytic semigroup in L^{p}(\R^N) if \alpha=2 and 1&lt;p&lt;\infty or \alpha&lt;2 and \frac{N}{N-\alpha}&lt; p&lt;\infty or \alpha&gt; 2 and \frac{N}{N-2}&lt;p&lt;\infty . The above bounds are shown to be sharp.

On bounded invertibility of a class of degenerate elliptic differential operators in Lp

A result on the existence of a bounded degenerated elliptic differential operator is proved and an a priori estimate in the space () is obtained.

High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m&lt;n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

A certain critical density property for invariant Harnack inequalities in H-type groups

We consider second order linear degenerate elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra of H-type. Extending a result by Gutiérrez and Tournier (2011) for the Heisenberg group, we prove a critical density estimate by assuming a condition of Cordes–Landis type. We then deduce an invariant Harnack inequality for the non-negative solutions from a result by Di Fazio, Gutiérrez, and Lanconelli (2008).

Weak Hopf lemma for the invariant Laplacian and related elliptic operators

We obtain a weak version of the Hopf lemma for the invariant Laplacian on the unit ball of the complex n-space. We also show that our result is sharp in some sense. Motivated by this result, we also consider a class of degenerate elliptic operators with the degeneracy depending on the distance to the boundary of the domain. We study the dependence of the validity of Hopf lemma on the degree of degeneracy of the operator. We show that Hopf lemma holds if the degeneracy is small and fails in general if the degeneracy is large. What is more interesting is the critical case for which we show that certain weak version of Hopf lemma holds.

Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Ω of ℝ N and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.

Existence and regularity of solutions to semi-linear Dirichlet problem of infinitely degenerate elliptic operators with singular potential term

In this paper, we study the Dirichlet problem for a class of semi-linear infinitely degenerate elliptic equations with singular potential term. By using the logarithmic Sobolev inequality and Hardy’s inequality, the existence and regularity of multiple nontrivial solutions have been proved.

Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators

In this paper we consider initial boundary value problem for semilinear parabolic equations involving strongly degenerate elliptic differential operators. Depending on the concrete types of nonlinearity we establish the existence of compact connected global attractors of semigroups generated by the problem under consideration.

The Harnack Inequality for a Class of Degenerate Elliptic Operators

We prove a Harnack inequality for distributional solutions to a type of degenerate elliptic partial differential equations in N dimensions. The differential operators in question are related to the Kolmogorov operator, made up of the Laplacian in the last N−1 variables, a first-order term corresponding to a shear flow in the direction of the first variable, and a bounded measurable potential term. The first-order coefficient is a smooth function of the last N−1 variables and its derivatives up to certain order do not vanish simultaneously at any point, making the operators in question hypoelliptic.

The solution of the Kato problem for degenerate elliptic operators with Gaussian bounds

We prove the Kato conjecture for degenerate elliptic operators on R n . More precisely, we consider the divergence form operator Lw = −w 1 divA∇, where w is a Muckenhoupt A2 weight and A is a complex-valued n × n matrix such that w 1 A is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup e tLw satisfies Gaussian bounds, then the weighted Kato square root estimate, kL 1/2 w f k L2(w) ≈ k∇ f k L2(w), holds.

Heat kernels for a class of degenerate elliptic operators using stochastic method

Formulas for heat kernels are found for degenerate elliptic operators by finding the probability density of the associated Ito diffusion. The formulas involve an integral of a product between a volume function and an exponential term.

Boundary blow-up solutions to degenerate elliptic equations with non-monotone inhomogeneous terms

Given a non-negative, and non-trivial continuous real-valued function h on Ω¯×[0,∞) such that h(x,0)=0 for all x∈Ω, we study the boundary value problem (BVP){Δ∞u=h(x,u)in Ωu=∞on ∂Ω, where Ω⊆RN,N≥2 is a bounded domain and Δ∞ is the ∞-Laplacian, a degenerate elliptic operator. In this paper, we investigate conditions on the inhomogeneous term h(x,t) under which Problem (BVP) admits a solution or fails to admit a solution in C(Ω). Some notable features of this work are that h(x,t) is not required to have any special structure, and no monotonicity condition is imposed on h(x,t). Furthermore, h(x,t) may be allowed to vanish in either of the variables.