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Weighted minimum $α$-Green energy problems

Natalia Zorii

TL;DR

This work addresses the weighted minimum $\alpha$-Green energy problem on a domain $D$ with an external field generated by a measure $\vartheta$, using the $\alpha$-Green kernel $g_\alpha$ and its perfectness properties. The author develops a Gauss variational framework, constructs a dual extremal problem, and characterizes minimizers $\lambda_{F,f}$ under various capacity and mass conditions, including explicit representations and convergence results as $F$ is approximated by compact or increasing/decreasing nets. Key contributions include solvability criteria in the finite and infinite capacity regimes, dual formulations, a detailed description of the minimizer's support, and strong continuity results for approximations, all grounded in the theory of $g_\alpha$-balayage and equilibrium measures. The results extend potential-theoretic methods to $\alpha\in(0,n]$, providing robust tools for energy minimization with external fields in non-compact settings and yielding convergence theorems crucial for approximating complex domains in applications.

Abstract

For the $α$-Green kernel $g^α_D$ on a domain $D\subset\mathbb R^n$, $n\geqslant2$, associated with the $α$-Riesz kernel $|x-y|^{α-n}$, where $α\in(0,n)$ and $α\leqslant2$, and a relatively closed set $F\subset D$, we investigate the problem on minimizing the Gauss functional \[\int g^α_D(x,y)\,d(μ\otimesμ)(x,y)-2\int g^α_D(x,y)\,d(\vartheta\otimesμ)(x,y),\] $\vartheta$ being a given positive (Radon) measure concentrated on $D\setminus F$, and $μ$ ranging over all probability measures of finite energy, supported in $D$ by $F$. For suitable $\vartheta$, we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when $F$ is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the $α$-Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).

Weighted minimum $α$-Green energy problems

TL;DR

This work addresses the weighted minimum -Green energy problem on a domain with an external field generated by a measure , using the -Green kernel and its perfectness properties. The author develops a Gauss variational framework, constructs a dual extremal problem, and characterizes minimizers under various capacity and mass conditions, including explicit representations and convergence results as is approximated by compact or increasing/decreasing nets. Key contributions include solvability criteria in the finite and infinite capacity regimes, dual formulations, a detailed description of the minimizer's support, and strong continuity results for approximations, all grounded in the theory of -balayage and equilibrium measures. The results extend potential-theoretic methods to , providing robust tools for energy minimization with external fields in non-compact settings and yielding convergence theorems crucial for approximating complex domains in applications.

Abstract

For the -Green kernel on a domain , , associated with the -Riesz kernel , where and , and a relatively closed set , we investigate the problem on minimizing the Gauss functional being a given positive (Radon) measure concentrated on , and ranging over all probability measures of finite energy, supported in by . For suitable , we find necessary and/or sufficient conditions for the existence of the solution to the problem, give a description of its support, provide various alternative characterizations, and prove convergence theorems when is approximated by partially ordered families of sets. The analysis performed is substantially based on the perfectness of the -Green kernel, discovered by Fuglede and Zorii (Ann. Acad. Sci. Fenn. Math., 2018).
Paper Structure (24 sections, 22 theorems, 134 equations)

This paper contains 24 sections, 22 theorems, 134 equations.

Table of Contents

  1. General conventions and preliminaries
  2. Basic concepts of the theory of potentials on a locally compact space
  3. $\alpha$-Riesz balayage
  4. $\alpha$-thinness of a set at infinity
  5. $\alpha$-harmonic measure on $\overline{\mathbb R^n}:=\mathbb R^n\cup\{\infty_{\mathbb R^n}\}$
  6. $\alpha$-Green kernel
  7. $\alpha$-Green balayage
  8. When does equality in (\ref{['bal4g']}) hold?
  9. The Gauss variational problem for $g_\alpha$-potentials
  10. Statement of the problem
  11. Main results
  12. Preparatory assertions
  13. A dual extremal problem
  14. Extremal measures
  15. Proof of Theorem \ref{['sol-1']}
  16. ...and 9 more sections

Key Result

Theorem 1.1

Given a closed set $A$ and a perfect kernel $\kappa$, assume $c_\kappa(A)<\infty$. Then for any $q\in(0,\infty)$, $\mathcal{E}^+_\kappa(A,q)$ is complete in the induced strong topology.

Theorems & Definitions (40)

  • Theorem 1.1
  • proof
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Corollary 1.6
  • Lemma 1.7
  • proof
  • ...and 30 more