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COMPARISON AND REGULARITY RESULTS FOR THE FRACTIONAL LAPLACIAN VIA SYMMETRIZATION METHODS
, 2012
"... In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use su ..."
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Cited by 5 (2 self)
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In this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional Laplacian. This allows to get sharp estimates for the solutions, obtained by comparing them with solutions of suitable radial problems. Furthermore, we use such result to prove a priori estimates for solutions in terms of the data, providing several regularity results which extend the well known ones for the classical Laplacian.
Symmetric Rearrangements Around Infinity with Applications to Lévy Processes
, 2011
"... We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger [FL76] and can be interpreted as involving symmetric rearrangements of domains around∞. As applications, we prove two comparison results for general Lévy processes and their ..."
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Cited by 3 (1 self)
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We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger [FL76] and can be interpreted as involving symmetric rearrangements of domains around∞. As applications, we prove two comparison results for general Lévy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Lévy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Lévy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi [PS11] for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Lévy process can only increase if the set and the Lévy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe [W83] for symmetric Lévy processes.
ON THE FIRST EIGENFUNCTION OF THE SYMMETRIC STABLE PROCESS IN A BOUNDED LIPSCHITZ DOMAIN
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