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In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x0 in D, then is D "almost'' a ball with center x0? This is the stability counterpart of the rigidity question (the statement above, without the two "almost'') for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x0.

By an easy “trick” taken from the caloric polynomial theory, we prove the existence of a basis of the Euclidean topology whose elements are resolutive sets of the heat equation. This result can be used to construct, with a very elementary approach, the Perron solution of the caloric Dirichlet problem on arbitrary bounded open subsets of the Euclidean space-time.

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at “$t=- \infty$” for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.

Let H be a linear second order partial differential operator with non-negative characteristic form in a strip S ⊂ R^N ×R. We assume that H as a fundamental solution, smooth out of its poles and bounded from above and from below by Gaussian kernels modeled on subriemannian doubling distances in R^N. Under these assumptions we show that H endows S with a structure of β-harmonic space. This allows us to study boundary value problems for L with a Perron-Wiener-Brelot-Bauer method, and to obtain pointwise regularity estimates at the boundary in terms of Wiener series modeled on the Gaussian kernels. Our analysis includes the proof of a scale invariant Harnack inequality for nonnegative solutions. We also show an application to the real hypersurphaces of C^{n+1} with given Levi-curvature.

Let L be a real sub-Laplacian on a stratified Lie group G. In this note we present some results on the Dirichlet problem for L with L^p-boundary data, on domains which are contractible with respect to the natural dilations of G. One of the main difficulties we overcome is the presence of non-regular boundary points for the usual Dirichlet problem for L. A potential theoretical approach is followed.

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In this note we present mean value characterizations of subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution ¡. These characterizations are based on suitable average operators on the level sets of ¡. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach and Reade. The results presented here generalize and carry forward former results of the authors in [6, 8].

We report on some Liouville-type theorems for a class of linear second-order partial differential equation with nonnegative characteristic form. The theorems we show improve our previous results.

We construct the fundamental solutions $\Gamma$ and $\gamma$ for the non-divergence form operators ${\textstyle\sum_{i,j}} a_{i,j}(x,t)X_iX_j-\partial_t$ and ${\textstyle\sum_{i,j}}a_{i,j}(x)X_iX_j$, where the $X_i$'s are Hörmander vector fields generating a stratified group $\mathbb{G}$ and $(a_{i,j})_{i,j}$ is a positive-definite matrix with Hölder continuous entries. We also provide Gaussian estimates of $\Gamma$ and its derivatives and some results for the relevant Cauchy problem. Suitable long-time estimates of $\Gamma$ allow us to construct $\gamma$ using both $t$-saturation and approximation arguments. [ABSTRACT FROM AUTHOR]

The maximum principle on a wide class of unbounded domains is proved for solutions to the partial differential inequality $\Delta_{\mathbb{G}}u+cu\geq 0$, where $c\leq 0$ and $\Delta_{\mathbb{G}}$ is a real sub-Laplacian. A potential theory approach is followed. [ABSTRACT FROM AUTHOR]