1,720,987 research outputs found
Large sets at infinity and Maximum Principle on unbounded domains for a class of sub-elliptic operators
Full text linkMaximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In this paper we consider a class of sub-elliptic operators L in RN and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian (by Deny, Hayman and Kennedy) and to the sub-Laplacians on stratified Lie groups (by Bonfiglioli and the second-named author)
Superparabolic Functions Related to Second Order Hypoelliptic Operators
No full textIn this paper, we consider a wide class of second order hypoelliptic partial differential operators with nonnegative characteristic form. We prove a monotone smoothing theorem and a representation formula for superparabolic functions
On Mean Value formulas for solutions to second order linear PDEs
Full text linkIn this paper we give a general proof of Mean Value formulas for solutions to second order linear PDEs, only based on the local properties of their fundamental solution Gamma. Our proof requires a kind of pointwise vanishing integral condition for the intrinsic gradient of Gamma. Combining our Mean Value formulas with a "descent method" due to Kuptsov, we obtain formulas with improved kernels. As an application, we implement our general results to heat operators on stratified Lie groups and to Kolmogorov operators
Stability of the mean value formula for harmonic functions in Lebesgue spaces
Full text linkLet D be an open subset of Rn with finite measure, and let x∈ D. We introduce the p-Gauss gap of D w.r.t. x to measure how far are the averages over D of the harmonic functions u∈ Lp(D) from u(x). We estimate from below this gap in terms of the ball gap of D w.r.t. x, i.e., the normalized Lebesgue measure of D B, being B the biggest ball centered at x contained in D. From these stability estimates of the mean value formula for harmonic functions in Lp-spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space W1,p′, where p′ is the conjugate exponent of p
ON THE CHARACTERIZATION OF THE HARMONIC PSEUDOSPHERES VIA KURAN’S FUNCTIONS AND SINGLE-LAYER POTENTIALS
No full textWe present some characterizations of the harmonic pseudospheres in terms of the so called Kuran’s functions and of the single-layer potentials. Our characterizations apply to solid, harmonically stable domains
On the Harmonic Characterization Of The Spheres: A Sharp Stability Inequality
No full textLet be a bounded open subset of with finite
-dimensional Hausdorff measure and let be a point of . We introduce a new harmonic invariant, that we call Kuran gap of w.r.t. . To define this new invariant, denoted , we use a family of harmonic functions introduced by \"Ulk\"u Kuran in \cite{kuran}. Our main stability result can be described as follows: if is sufficiently regular just in one of the points of nearest to , then is bounded from below by a kind of isoperimetric index, precisely the normalized difference between and , being the biggest ball contained in and centered at .
This partially extends and improves a stability result by Preiss and Toro.
By our stability result, we also obtain new rigidity results: (i) a characterization of the Euclidean spheres in terms of single-layer potentials, improving previous theorems by Fichera and by Shahgholian; (ii) a sufficient condition for a harmonic pseudosphere to be a Euclidean sphere, partially extending and improving rigidity results by Lewis and Vogel
On the Harmonic characterization of domains via mean value formulas
Full text linkThe Euclidean ball have the following harmonic characterization, via Gauss-mean value property: Let D be an open set with finite Lebesgue measure and let x0 be a point of D. If
for every nonnegative harmonic function u in D, then D is a Euclidean ball centered at x0. On the other hand, on every sufficiently smooth domain D and for every point x0 in D there exist Radon measures μ such thatfor every nonnegative harmonic function u in D. In this paper we give sufficient conditions so that this last mean value property characterizes the domain D
Liouville Theorems and Detours, Nonlinear Analysis: Theory, Methods & Applications
No full textAtti di Convegn
A sharp stability result for the Gauss mean value formula
Full text linkWe prove a quantitative stability result for the Gauss mean value formula. We also show by an example that the estimate proved here is sharp
Erratum to: Global Lp estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients : Lp estimates for degenerate Ornstein-Uhlenbeck type operators (Mathematische Nachrichten, (2013), 286, 11-12, (1087-1101), 10.1002/mana.201200189)
Full text linkIn this note we point out and correct a mistake in our paper “Global Lp estimates for degenerate Ornstein–Uhlenbeck operators with variable coefficients”, published in Math. Nachr. 286 (2013), no. 11–12, 1087–1101
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