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Register a new userA nonlocal isoperimetric problem with dipolar repulsion
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We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order $Gamma$-limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.
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We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.
In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.
Conditions for the existence and uniqueness of weak solutions for a class of nonlinear nonlocal degenerate parabolic equations are established. The asymptotic behaviour of the solutions as time tends to infinity are also studied. In particular, the finite time extinction and polynomial decay properties are proved.
In this paper, we proceed to study the nonlocal diffusion problem proposed by Li and Wang [8], where the left boundary is fixed, while the right boundary is a nonlocal free boundary. We first give some accurate estimates on the longtime behavior by constructing lower solutions, and then investigate the limiting profiles of this problem when the expanding coefficient of free boundary converges to $0$ and $yy$, respectively. At last, we focus on two important kinds of kernel functions, one of which is compactly supported and the other behaves like $|x|^{-gamma}$ with $gammain(1,2]$ near infinity. With the help of some upper and lower solutions, we obtain some sharp estimates on the longtime behavior and rate of accelerated spreading.
The aim of this paper is to deal with the elliptic pdes involving a nonlinear integrodifferential operator, which are possibly degenerate and covers the case of fractional $p$-Laplacian operator. We prove the existence of a solution in the weak sense to the problem begin{align*} begin{split} -mathscr{L}_Phi u & = lambda |u|^{q-2}u,,mbox{in},,Omega, u & = 0,, mbox{in},, mathbb{R}^Nsetminus Omega end{split} end{align*} if and only if a weak solution to begin{align*} begin{split} -mathscr{L}_Phi u & = lambda |u|^{q-2}u +f,,,,fin L^{p}(Omega), u & = 0,, mbox{on},, mathbb{R}^Nsetminus Omega end{split} end{align*} ($p$ being the conjugate of $p$), exists in a weak sense, for $qin(p, p_s^*)$ under certain condition on $lambda$, where $-mathscr{L}_Phi $ is a general nonlocal integrodifferential operator of order $sin(0,1)$ and $p_s^*$ is the fractional Sobolev conjugate of $p$. We further prove the existence of a measure $mu^{*}$ corresponding to which a weak solution exists to the problem begin{align*} begin{split} -mathscr{L}_Phi u & = lambda |u|^{q-2}u +mu^*,,,mbox{in},, Omega, u & = 0,,, mbox{in},,mathbb{R}^Nsetminus Omega end{split} end{align*} depending upon the capacity.
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