Towards an L p-potential theory for sub-Markovian semigroups: variational inequalities and balayage theory

Abstract

We give a new variational approach toL p -potential theory for sub-Markovian semigroups. It is based on the observation that the Gâteaux-derivative of the corresponding L p-energy functional is a monotone operator. This allows to apply the well established theory of Browder and Minty on monotone operators to the nonlinear problems in L p-potential theory. In particular, using this approach it is possible to avoid any symmetry assumptions of the underlying semigroup. We prove existence of corresponding (r, p)-equilibrium potentials and obtain a complete characterization in terms of a variational inequality. Moreover we investigate associated potentials and encounter a natural interpretation of the so-called nonlinear potential operator in the context of monotone operators.

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Correspondence to Walter Hoh or Niels Jacob.

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Hoh, W., Jacob, . Towards an L p-potential theory for sub-Markovian semigroups: variational inequalities and balayage theory. J.evol.equ. 4, 297–312 (2004). https://doi.org/10.1007/s00028-003-0145-4

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Keywords

  • Variational Inequality
  • Nonlinear Problem
  • Variational Approach
  • Monotone Operator
  • Complete Characterization