We establish an upper bound of the sum of the eigenvalues for the Dirichlet problem of the fractional Laplacian. Our result is obtained by a subtle computation of the Rayleigh quotient for specific functions.
In the whole space $mathbb R^d$, linear estimates for heat semi-group in Besov spaces are well established, which are estimates of $L^p$-$L^q$ type, maximal regularity, e.t.c. This paper is concerned with such estimates for semi-group generated by th
e Dirichlet Laplacian of fractional order in terms of the Besov spaces on an arbitrary open set of $mathbb R^d$.
In this paper, we show the existence of a sequence of eigenvalues for a Dirichlet problem involving two mixed fractional operators with different orders. We provide lower and upper bounds for the sum of the eigenvalues. Applications of mixed fraction
al operators with different orders include medicine, plasma physics, and population dynamics.
We establish Ambrosetti--Prodi type results for viscosity and classical solutions of nonlinear Dirichlet problems for the fractional Laplace and comparable operators. In the choice of nonlinearities we consider semi-linear and super-linear growth cas
es separately. We develop a new technique using a functional integration-based approach, which is more robust in the non-local context than a purely analytic treatment.
We prove the long-standing Payne conjecture that the $k^{text{th}}$ eigenvalue in the buckling problem for a clamped plate is not less than the ${k+1}^{text{st}}$ eigenvalue for the membrane of the same shape which is fixed on the boundary. Moreover,
we show that the Payne conjecture is still true for $n$-dimensional case ($nge 2)$.
We study the bilinear estimates in the Sobolev spaces with the Dirichlet and the Neumann boundary condition. The optimal regularity is revealed to get such estimates in the half space case, which is related to not only smoothness of functions and but
also boundary behavior. The crucial point for the proof is how to handle boundary values of functions and their derivatives.
Ying Wang
,Hongxing Chen
,Hichem Hajaiej
.
(2020)
.
"Krogers upper bound types for the Dirichlet eigenvalues of the fractional Laplacian"
.
Hichem Hajaiej
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