An asymptotic expansion for the fractional \(p\)-Laplacian and for gradient-dependent nonlocal operators. (English) Zbl 07517068
Summary: Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well-known equivalence between harmonic functions and mean value properties. In the nonlocal setting of fractional harmonic functions, such an equivalence still holds, and many applications are nowadays available. The nonlinear case, corresponding to the \(p\)-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the nonlocal, nonlinear mean value kernel, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) \(p\)-Laplacian (for \(p\geq 2)\) and to other gradient-dependent nonlocal operators.
MSC:
| 47-XX | Operator theory |
| 35-XX | Partial differential equations |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 28D20 | Entropy and other invariants |
| 82B10 | Quantum equilibrium statistical mechanics (general) |
Keywords:
mean value formulas; fractional \(p\)-Laplacian; gradient-dependent operators; nonlocal \(p\)-Laplacian; infinite fractional LaplacianReferences:
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