### A characterization of caloric morphisms between manifolds.

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This paper is devoted to a study of harmonic mappings $\phi $ of a harmonic space $\tilde{E}$ on a harmonic space $E$ which are related to a family of harmonic mappings of $\tilde{E}$ into $\tilde{E}$. In this way balayage in $E$ may be reduced to balayage in $E$. In particular, a subset $A$ of $E$ is polar if and only if ${\phi}^{-1}\left(A\right)$ is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.

We derive estimates for various quantities which are of interest in the analysis of the Ginzburg-Landau equation, and which we bound in terms of the $GL$-energy ${E}_{\epsilon}$ and the parameter $\epsilon $. These estimates are local in nature, and in particular independent of any boundary condition. Most of them improve and extend earlier results on the subject.

We consider the general Schrödinger operator $L=div\left(A\left(x\right){\nabla}_{x}\right)-\mu $ on a half-space in ℝⁿ, n ≥ 3. We prove that the L-Green function G exists and is comparable to the Laplace-Green function ${G}_{\Delta}$ provided that μ is in some class of signed Radon measures. The result extends the one proved on the half-plane in [9] and covers the case of Schrödinger operators with potentials in the Kato class at infinity $K{\u2099}^{\infty}$ considered by Zhao and Pinchover. As an application we study the cone ${}_{L}\left(\mathbb{R}\u207f\u208a\right)$ of all positive L-solutions continuously vanishing...

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type ${L}_{1}u=-{\mu}_{1}v$, ${L}_{2}v=-{\mu}_{2}u$, on a domain $D$ of ${\mathbb{R}}^{d}$, where ${\mu}_{1}$ and ${\mu}_{2}$ are suitable measures on $D$, and ${L}_{1}$, ${L}_{2}$ are two second order linear differential elliptic operators on $D$ with coefficients of class ${\mathcal{C}}^{\infty}$. We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with ${L}_{1}$ and ${L}_{2}$, and a convergence...