In this paper we study a variational Neumann problem for the higher order $s$-fractional Laplacian, with $s>1$. In the process, we introduce some non-local Neumann boundary conditions that appear in a natural way from a Gauss-like integration formula.
We consider the parabolic polyharmonic diffusion and $L^2$-gradient flows of the $m$-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the
curvature of the initial curve is small in $L^2$, then the evolving curve converges exponentially in the $C^infty$ topology to a straight horizontal line segment. The same behaviour is shown for the $L^2$-gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on $m$.
We present a construction of harmonic functions on bounded domains for the spectral fractional Laplacian operator and we classify them in terms of their divergent profile at the boundary. This is used to establish and solve boundary value problems as
sociated with nonhomogeneous boundary conditions. We provide a weak-$L^1$ theory to show how problems with measure data at the boundary and inside the domain are well-posed. We study linear and semilinear problems, performing a sub- and supersolution method, and we finally show the existence of large solutions for some power-like nonlinearities.
We study the regularity up to the boundary of solutions to the Neumann problem for the fractional Laplacian. We prove that if $u$ is a weak solution of $(-Delta)^s u=f$ in $Omega$, $mathcal N_s u=0$ in $Omega^c$, then $u$ is $C^alpha$ up tp the bound
ary for some $alpha>0$. Moreover, in case $s>frac12$, we then show that $uin C^{2s-1+alpha}(overlineOmega)$. To prove these results we need, among other things, a delicate Moser iteration on the boundary with some logarithmic corrections. Our methods allow us to treat as well the Neumann problem for the regional fractional Laplacian, and we establish the same boundary regularity result. Prior to our results, the interior regularity for these Neumann problems was well understood, but near the boundary even the continuity of solutions was open.
We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then d
iscuss some properties of the fractional Poissons equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.
The fractional Leibniz rule is generalized by the Coifman-Meyer estimate. It is shown that the arbitrary redistribution of fractional derivatives for higher order with the corresponding correction terms.
B. Barrios
,L. Montoro
,I. Peral
.
(2018)
.
"Neumann conditions for the higher order $s$-fractional Laplacian $(-Delta)^su$ with $s>1$"
.
Bego\\~na Barrios
هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا