Nonlinear Analysis-Theory Methods & Applications最新文献

We provide global a priori second derivative $L^{\delta }$-estimates for a class of singular fully nonlinear elliptic equations with right hand side terms of $L^n$.

We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the $q$-Laplacian.

In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.

In this paper, we prove a Talenti-type comparison theorem for the $p$-Laplacian with Dirichlet boundary conditions on open subsets of a normalized $\mathrm{RCD}(K,N)$ space with $K>0$ and $N\in (1,\infty )$. The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the $p$-Laplacian and a related quantitative stability result.

We study a non-local evolution equation on the hyperbolic space $H^N$. We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions.

We also consider a non-local, non-linear convection–diffusion equation on $H^N$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain–Brezis–Mironescu.

We consider heat operators on a bounded domain $\Omega \subseteq R^n$, with a critically singular potential diverging as the inverse square of the distance to $\partial \Omega$. Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) $\Omega$ was convex, (ii) the control must be prescribed along all of $\partial \Omega$, and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of $\Omega$, (ii) allow for the control to be localized near any $x_0\in \partial \Omega$, and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for $\partial \Omega$ and the lower-order coefficients. The key novelty is a local Carleman estimate near $x_0$, with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of $\partial \Omega$.

In this paper we prove existence and regularity of weak solutions for the following system $\left(\begin{array}{c}-\text{div}(M\left(x\right)\nabla u)+g(x,u,v)=f\text{in}\Omega \\ -\text{div}(M\left(x\right)\nabla v)=h(x,u,v)\text{in}\Omega \\ u=v=0\text{on}\partial \Omega ,\end{array}\right)$where $\Omega$ is an open bounded subset of $R^N$, for $N>2$, $f\in L^m\left(\Omega \right)$, $M$ is a matrix with Lipschitz coefficients, $m>1$ and $g$, $h$ are two Carathéodory functions. We prove that under appropriate conditions on $g$ and $h$, there exist solutions which escape the predicted regularity by the classical Stampacchia’s theory causing the so-called regularizing effect.

In this paper, we study generalized weighted Ricci curvatures, which include the $N$-Ricci curvature and the projective Ricci curvature with totally different geometric meanings. We completely classify a class of weakly weighted Einstein-Finsler metrics.

We study Hardy inequalities for antisymmetric functions in three different settings: Euclidean space, torus and the integer lattice. In particular, we show that under the antisymmetric condition the sharp constant in Hardy inequality increases substantially and grows as $d^4$ as $d\to \infty$ in all cases. As a side product, we prove Hardy inequality on a domain whose boundary forms a corner at the point of singularity $x=0$.

We study the wellposedness, stabilization and blow up of solutions of the wave equation with nonlinearities of arbitrary growth and locally distributed nonlinear dissipation posed in a 2-dimensional compact Riemannian manifold $(M,g)$ without boundary. Differently of the previous literature we give a different proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof, even in the case of subcritical, critical or super critical growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than $d$ of the ground state, where $d$ is the level of the Mountain Pass Theorem.