Nonlinear Analysis-Theory Methods & Applications最新文献

We study the regularity of local minimisers of a prototypical free-discontinuity problem involving both a manifold-valued constraint on the maps (which are defined on a bounded domain $\Omega \subset R^2$) and a variable-exponent growth in the energy functional. To this purpose, we first extend to this setting the Sobolev approximation result for special function of bounded variation with small jump set originally proved by Conti, Focardi, and Iurlano (Conti et al., 2017; Conti et al., 2019) for special functions of bounded deformation. Secondly, we use this extension to prove regularity of local minimisers.

In this paper we introduce two new fractional versions of the Laplacian. The first one is based on the classical formula that writes the usual Laplacian as the sum of the eigenvalues of the Hessian. The second one comes from looking at the classical fractional Laplacian as the mean value (in the sphere) of the 1-dimensional fractional Laplacians in lines with directions in the sphere. To obtain this second new fractional operator we just replace the mean value by the mid-range of 1-dimensional fractional Laplacians with directions in the sphere. For these two new fractional operators we prove a comparison principle for viscosity sub and supersolutions and then we obtain existence and uniqueness for the Dirichlet problem, that turns out to be nonlinear. Strong maximum and comparison principles also hold. Finally, we prove that for the first operator we recover the classical Laplacian in the limit as $s\nearrow 1$, while for the second operator we obtain the sum of the smallest and the largest classical Hessian eigenvalues.

We obtain Serrin-type theorems of the solution to overdetermined problems in a warped product manifold with a nonconstant Neumann boundary condition by applying the maximum principle to suitable subharmonic functions and integral identities.

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form $f(x,u)$ under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension $n\in N$.

The Jordan–Moore–Gibson–Thompson equation $\tau u_{ttt}+\alpha u_{tt}=\beta \Delta u_t+\gamma \Delta u+{\left(f\left(u\right)\right)}_{tt}$is considered in a smoothly bounded domain $\Omega \subset R^n$ with $n\le 3$, where $\tau >0,\beta >0,\gamma >0$, and $\alpha \in R$.

Firstly, it is seen that under the assumption that $f\in C^3\left(R\right)$ is such that $f\left(0\right)=0$, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial–boundary value problem admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is such that $\text{if}{T}_{max}<\infty ,\text{then}\underset{t\nearrow T_{max}}{lim\; sup}{\Vert u(\cdot ,t)\Vert }_{L^{\infty }\left(\Omega \right)}=\infty .(\star )$This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that