L-harmonic Functions Research Articles

Gaussian estimates vs. elliptic regularity on open sets

AbstractGiven an elliptic operator $$L= - {{\,\textrm{div}\,}}(A \nabla \cdot )$$ L = - div ( A ∇ · ) subject to mixed boundary conditions on an open subset of $$\mathbb {R}^d$$ R d , we study the relation between Gaussian pointwise estimates for the kernel of the associated heat semigroup, Hölder continuity of L-harmonic functions and the growth of the Dirichlet energy. To this end, we generalize an equivalence theorem of Auscher and Tchamitchian to the case of mixed boundary conditions and to open sets far beyond Lipschitz domains. Yet, we prove the consistency of our abstract result by encompassing operators with real-valued coefficients and their small complex perturbations into one of the aforementioned equivalent properties. The resulting kernel bounds open the door for developing a harmonic analysis for the associated semigroups on rough open sets.

Non-linear operators of divergence form on the Sierpinski gasket

We consider non-linear operators L of divergence type on the Sierpinski gasket. We prove the Harnack inequality for non-negative L-harmonic functions on the Sierpinski gasket. The proof is based on Moser’s method, and we construct a new measure λ by combining a Hausdorff measure and an energy measure, which plays an essential role at key steps. The Harnack inequality is achieved by involving the Caccioppoli type inequality and the weighted Sobolev inequality which give the local boundedness and the Weak Harnack inequality of solutions with respect to the measure λ.

Isomorphisms of variable Hardy spaces associated with Schrödinger operators

Let L:= −Δ + V be the Schrodinger operator on ℝn with n ≥ 3, where V is a non-negative potential satisfying Δ−1 (V) ∈ L∞(ℝn). Let w be an L-harmonic function, determined by V, satisfying that there exists a positive constant o such that, for any x ∈ ℝn, 0 < δ ≤ w(x) ≤ 1. Assume that p(·): ℝn → (0, 1] is a variable exponent satisfying the globally log-Holder continuous condition. In this article, the authors show that the mappings $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto wf \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ and $$H_L^{p\left( \cdot \right)}\left( {{\mathbb{R}^n}} \right)f \mapsto {\left( { - {\rm{\Delta }}} \right)^{1/2}}{L^{ - 1/2}}\left( f \right) \in {H^{p\left( \cdot \right)}}\left( {{\mathbb{R}^n}} \right)$$ are isomorphisms between the variable Hardy spaces (ℝn), associated with L, and the variable Hardy spaces Hp(·)(ℝn).

Counting dimensions of L-harmonic functions with exponential growth

Let $$\Omega \subset {\mathbb {R}}^{n-1}$$ be a bounded open set, $$X=\Omega \times {\mathbb {R}}\subseteq {\mathbb {R}}^{n}$$ be the infinite strip. Let L be a second order uniformly elliptic operator of divergence form acting on a function $$f\in W_{\text {loc}}^{1,2}(X)$$ given by $$Lf=\sum _{i,j=1}^{n}\frac{\partial }{\partial x_{i}}\bigl (a^{ij}(x)\frac{\partial f}{\partial x_{j}}\bigr )$$. It is natural to consider the solutions of $$Lu=0$$ with boundary value $$u|_{\partial \Omega \times {\mathbb {R}}}=0$$ and exponential growth at most d: $$|u(x',x_{n})|\le {\tilde{C}}e^{d|x_{n}|}$$ for some $${\tilde{C}}>0$$. Denote by $${\mathcal {A}}_{d}$$ the solution space. In (Acta Math Sin (Engl Ser)15:525–534, 1999), Hang and Lin proved that $$\text {dim}{\mathcal {A}}_{d}\le Cd^{n-1}$$. The power $$n-1$$ is sharp, but one may wonder whether there are more precise estimates for the constant C. In this note, we consider some natural subspaces of $${\mathcal {A}}_{d}$$ and obtain some estimates of dimensions of these subspaces. Compared with the case $$L=\Delta _{X}$$, when d is sufficiently large, the estimates obtained in this note are sharp both on the power $$n-1$$ and the constant C.

Isomorphisms and several characterizations of Musielak-Orlicz-Hardy spaces associated with some Schrödinger operators

Let L ≔ −Δ + V be a Schrodinger operator on ℝ n with n ⩾ 3 and V ⩾ 0 satisfying Δ−1 V ∈ L ∞(ℝ n ). Assume that φ: ℝ n × [0,∞) → [0,∞) is a function such that φ(x, ·) is an Orlicz function, φ(·, t) ∈ A ∞(ℝ n ) (the class of uniformly Muckenhoupt weights). Let w be an L-harmonic function on ℝ n with 0 < C 1 ⩽ w ⩽ C 2, where C 1 and C 2 are positive constants. In this article, the author proves that the mapping $$H_{\phi ,L} (\mathbb{R}^n ) \mathrel\backepsilon f \mapsto wf \in H_\phi (\mathbb{R}^n )$$ is an isomorphism from the Musielak-Orlicz-Hardy space associated with $$L,H_{\phi ,L} (\mathbb{R}^n )$$ , to the Musielak-Orlicz-Hardy space $$H_\phi (\mathbb{R}^n )$$ under some assumptions on φ. As applications, the author further obtains the atomic and molecular characterizations of the space $$H_{\phi ,L} (\mathbb{R}^n )$$ associated with w, and proves that the operator $${( - \Delta )^{ - 1/2}}{L^{1/2}}$$ is an isomorphism of the spaces $$H_{\phi ,L} (\mathbb{R}^n )$$ and $$H_\phi (\mathbb{R}^n )$$ . All these results are new even when φ(x, t) ≔ t p , for all x ∈ ℝ n and t ∈ [0,∞), with p ∞ (n/(n + μ0), 1) and some μ0 ∈ (0, 1].

О взаимосвязи разрешимостей некоторых краевых задач для L-гармонических функций в неограниченных областях римановых многообразий

О взаимосвязи разрешимостей некоторых краевых задач для L-гармонических функций в неограниченных областях римановых многообразий

On characterization of Poisson integrals of Schrödinger operators with BMO traces

Let L be a Schrödinger operator of the form L=−Δ+V acting on L2(Rn) where the nonnegative potential V belongs to the reverse Hölder class Bq for some q⩾n. Let BMOL(Rn) denote the BMO space on Rn associated to the Schrödinger operator L. In this article we will show that a function f∈BMOL(Rn) is the trace of the solution of Lu=−utt+Lu=0, u(x,0)=f(x), where u satisfies a Carleson conditionsupxB,rBrB−n∫0rB∫B(xB,rB)t|∇u(x,t)|2dxdt⩽C<∞. Conversely, this Carleson condition characterizes all the L-harmonic functions whose traces belong to the space BMOL(Rn). This result extends the analogous characterization founded by Fabes, Johnson and Neri in [13] for the classical BMO space of John and Nirenberg.

Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $$L=\Delta ^{\alpha /2}+ b\cdot \nabla $$ with $$\alpha \in (1,2)$$ . We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on $$\mathcal{C }^{1,1}$$ bounded open sets.

The Dirichlet problem and the inverse mean‐value theorem for a class of divergence form operators

The aim of this paper is to study some classes of second-order divergence-form partial differential operators ℒ of sub-Riemannian type. Our main assumption is the C∞-hypoellipticity of ℒ, together with the existence of a well-behaved fundamental solution Γ(x, y) for ℒ. We consider the mean-integral operator Mr naturally associated to the mean-value theorem for the ℒ-harmonic functions and we investigate the following topics: the positivity set of the kernel associated to Mr; the role of Mr in solving the homogeneous Dirichlet problem related to ℒ in the Perron–Wiener–Brelot sense; the existence of an inverse mean-value theorem characterizing the sub-Riemannian ‘balls’ Ωr(x), superlevel sets of Γ(x,·). This last result extends a previous theorem by Kuran [Bull. London Math. Soc. 1972]. As side-results, we provide a short proof of the Strong Maximum Principle for ℒ using Mr, a Poisson–Jensen formula for the ℒ-subharmonic functions and several results concerning the geometry of the sets Ωr(x).

Generalized harmonic functions of Riemannian manifolds with ends

We study L-harmonic functions (solutions of the stationary Schrodinger equation) on arbitrary noncompact Riemannian manifolds with finitely many ends. We establish some existence and uniqueness results, and obtain sharp dimension estimates for L-harmonic functions on such manifolds.

L-harmonic functions with polynomial growth of a fixed rate

Yau made the following conjecture: For a complete noncompact manifold with nonnegative Ricci curvature the space of harmonic functions with polynomial growth of a fixed rate is finite dimensional. we extend the result on the Laplace operator to that on the symmetric diffusion operator, and prove the space of L-harmonic functions with polynomial growth of a fixed rate is finite-dimensional, when m-dimensional Bakery-Emery Ricci curvature of the symmetric diffusion operator on the complete noncompact Riemannian manifold is nonnegative.

Jump processes, ℒ-harmonic functions, continuity estimates and the Feller property

Soit ν={ν(x, ⋅)}x∈ℝd une famille de mesures de Lévy, ce travail étudie la régularité de fonctions harmoniques et la propriété de Feller du processus de saut correspondant. Le but principal est d’établir des estimations de continuité pour les fonctions harmoniques sous des conditions faibles sur la famille ν. À la différence des contributions précédentes cette méthode couvre des cas où les bornes inférieures de la probabilité d’atteindre de petits ensembles dégénèrent.

Harnack inequality and Liouville properties for L-harmonic operator

For any complete manifold with nonnegative Bakry–Emery's Ricci curvature, we prove the gradient estimate of L-harmonic function. As application, we use this gradient estimate to deduce the localized version of the Harnack inequality for L-harmonic operator and some Liouville properties of positive or bounded L-harmonic function.

Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds

Let L = Δ − ∇ ϕ ⋅ ∇ be a symmetric diffusion operator with an invariant measure μ ( d x ) = e − ϕ ( x ) d x on a complete non-compact Riemannian manifold M. We give the optimal conditions on “the m-dimensional Ricci curvature associated with L” so that various Liouville theorems hold for L-harmonic functions, and that the heat semigroup P t = e t L has the C 0 -diffusion property and is unique in L 1 ( M , μ ) . As applications, we give the optimal conditions for the uniqueness of the positive L-invariant measure and the L 1 -uniqueness of the intrinsic Schrödinger operators on complete non-compact Riemannian manifolds. We also give a criterion for the finiteness of the total mass of the L-invariant measure and establish the Calabi–Yau volume growth theorem for the L-invariant measure on complete Riemannian manifolds on which “the m-dimensional Ricci curvature associated with L” is non-negative. This leads us to prove that if M is a complete Riemannian manifold with a finite L-invariant measure for which the associated m-dimensional Ricci curvature is non-negative, then M is compact. Moreover, we obtain an upper bound diameter estimate of such Riemannian manifolds by using the dimension of L, the total μ-volume of M and the upper bound of the μ-volume of geodesic balls of a fixed radius. Finally, using the variational formulae in Riemannian geometry, we give a new proof of the Bakry–Qian generalized Laplacian comparison theorem.

Some results and examples about the behavior of harmonic functions and Green’s functions with respect to second order elliptic operators

Let M be a manifold and let L be a sufficiently smooth second order elliptic operator in M such that (M, L) is a transient pair. It is first shown that if L is symmetric with respect to some density in M, there exists a positive L-harmonic function in M which dominates L-Green’s function at infinity. Other classes of elliptic operators are investigated and examples are constructed showing that this property may fail if the symmetry assumption is removed. Another part of the paper deals with the existence of critical points for certain L-harmonic functions with periodicity properties. A class of small perturbations of second order elliptic operators is also described.

Counting Dimensions of L-Harmonic Functions

In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x_1,...,x_n}, a second order uniformly elliptic operator of divergence form, L, acting on a function f in H^1_loc(R^n) is given by Lf = sum_{ij} d/dx_i (a^{ij}(x) df/dx_j) where (a^{ij}(x)) is an n x n symmetric matrix satisfying the ellipticity bounds \lambda I <= (a^{ij}) <= Lambda I for some constants 0 < lambda <= Lambda < \infty. Other than the ellipticity bounds, we only assume that the coefficients (a_{ij}) are merely measurable functions.

Divergence Form Operators on Fractal-like Domains

We consider elliptic operators L in divergence form on certain domains in Rd with fractal volume growth. The domains we look at are pre-Sierpinski carpets, which are derived from higher dimensional Sierpinski carpets. We prove a Harnack inequality for non-negative L-harmonic functions on these domains and establish upper and lower bounds for the corresponding heat equation.

A UNIFORM GROWTH ESTIMATES OF SOLUTIONS OF ASYMPTOTICALLY ELLIPTIC OPERATORS

This paper introduces a generic eigenvalue flow of a parameter family of operators, where the corresponding eigenfunction is continuous in parameters. Then the author applies the result to the study of polynomial growth L-harmonic functions. Under the assumption that the operator has some weakly conic structures at infinity which is not necessarily unique, a Harnack type uniform growth estimate is obtained.

Solutions of Semilinear Differential Equations Related to Harmonic Functions

This is an attempt to establish a link between positive solutions of semilinear equations Lu=−ψ(u) and Lv=ψ(v) where L is a second order elliptic differential operator and ψ is a positive function. The equations were investigated separately by a number of authors. We try to link them via positive solutions of a linear equation Lu=0 (we call them L-harmonic functions). Let D be an arbitrary open subset of Rd and let U(D), V(D) and H(D) stand for the sets of all positive solutions in D for three equations mentioned above. We establish a 1–1 correspondence between certain subclasses of these classes. Similar results are obtained also for the corresponding parabolic equations. A probabilistic interpretation in terms of a superdiffusion is given in [1].

Trace on the boundary for solutions of nonlinear differential equations

Let L be a second order elliptic differential operator in Rd with no zero order terms and let E be a bounded domain in Rd with smooth boundary 'E. We say that a function h is L-harmonic if Lh = 0 in E. Every positive L-harmonic function has a unique representation h(x) J(x, y)i(dy), E where k is the Poisson kernel for L and z' is a finite measure on &E. We call z the trace of h on ME. Our objective is to investigate positive solutions of a nonlinear equation Lu = u' in E for 1 2]. We associate with every solution u a pair (IF, i), where F is a closed subset of &E and zis a Radon measure on 0 -E F. We call (F, zi) the trace of u on ME. F is empty if and only if u is dominated by an L-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions for a pair (F, z') to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.