This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration compariso...This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.展开更多
We establish Holder continuity of the horizontal gradient of weak solu-tions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg Group.
We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial da...We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation.The phenomenon is a priori nontrivial due to the nonlocal structure of the equation.Our approach is based on Kato’s method using Picard’s interation,which can be apdated to the multi-dimensional case and other nonlinear non-local equations.We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.展开更多
基金supported by the ANR projects“HAB”and“NONLOCAL”,the Spanish Research Project MTM2011-24696the INDAM-GNAMPA Project 2014“Analisi qualitativa di soluzioni di equazioni ellittiche e di evoluzione”(Italy)
文摘This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of R^N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.
基金S.Mukherjee was partially supported by“Variationaaliset integraalit geometr”and GHAIA Marie Skłodowska-Curie grant agreement No.777822 under Horizon 2020.
文摘We establish Holder continuity of the horizontal gradient of weak solu-tions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg Group.
基金supported by the Simons Foundation,grant#354889。
文摘We establish local and global well-posedness of the 2D dissipative quasigeostrophic equation in critical mixed norm Lebesgue spaces.The result demonstrates the persistence of the anisotropic behavior of the initial data under the evolution of the 2D dissipative quasi-geostrophic equation.The phenomenon is a priori nontrivial due to the nonlocal structure of the equation.Our approach is based on Kato’s method using Picard’s interation,which can be apdated to the multi-dimensional case and other nonlinear non-local equations.We develop time decay estimates for solutions of fractional heat equation in mixed norm Lebesgue spaces that could be useful for other problems.
