Nonlinear Analysis-Theory Methods & Applications最新文献

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\overline{M}=I{\times }_{\lambda }M$. Based on the conjecture proposed by Ren–Wang in Ren and Wang (2020), which is valid for $k\ge n-2$, we derive curvature estimates for equation ${\sigma }_k\left(\kappa \right)=\psi (V,\nu \left(V\right))$ through a straightforward proof. Furthermore, we also obtain an existence result for the star-shaped compact hypersurface $\Sigma$ satisfying the above equation by the degree theory under some sufficient conditions.

We consider a three-dimensional micromagnetic model with Dzyaloshinskii-Moriya interaction in a thin-film regime for boundary vortices. In this regime, we prove a dimension reduction result: the nonlocal three-dimensional model reduces to a local two-dimensional Ginzburg–Landau type model in terms of the averaged magnetisation in the thickness of the film. This reduced model captures the interaction between boundary vortices (so-called renormalised energy), that we determine by a $\Gamma$-convergence result at the second order and then we analyse its minimisers. They nucleate two boundary vortices whose position depends on the Dzyaloshinskii-Moriya interaction.

This work revolves around properties and applications of functions whose nonlocal gradient, or more precisely, finite-horizon fractional gradient, vanishes. Surprisingly, in contrast to the classical local theory, we show that this class forms an infinite-dimensional vector space. Our main result characterizes the functions with zero nonlocal gradient in terms of two simple features, namely, their values in a layer around the boundary and their average. The proof exploits recent progress in the solution theory of boundary-value problems with pseudo-differential operators. We complement these findings with a discussion of the regularity properties of such functions and give illustrative examples. Regarding applications, we provide several useful technical tools for working with nonlocal Sobolev spaces when the common complementary-value conditions are dropped. Among these, are new nonlocal Poincaré inequalities and compactness statements, which are obtained after factoring out functions with vanishing nonlocal gradient. Following a variational approach, we exploit the previous findings to study a class of nonlocal partial differential equations subject to natural boundary conditions, in particular, nonlocal Neumann-type problems. Our analysis includes a proof of well-posedness and a rigorous link with their classical local counterparts via $\Gamma$-convergence as the fractional parameter tends to 1.

We consider manifold constrained weak solutions of quasilinear uniformly elliptic systems of divergence type with a source term that grows at most quadratically with respect to the gradient of the solution. As we impose that the solution lies on a Riemannian manifold, the classical smallness condition for regularity can be relaxed to an inequality relating strict convexity of the squared distance and growth of the leading order term in the tangent component of the source. As a key tool for the proof of a partial regularity result, we derive a fully intrinsic Caccioppoli inequality which may be of independent interest. Finally we show how the systems under consideration have a variational nature and arise in the context of $F$- or $V$-harmonic maps.

We consider on Riemannian manifolds the nonlinear evolution equation ${\partial }_{t}u={\Delta }_p\left(u^{1/(p-1)}\right),$where $p>1$. This equation is also known as a doubly non-linear parabolic equation or Trudinger’s equation. We prove that weak subsolutions of this equation have a sub-Gaussian upper bound and prove that this upper bound is sharp for a specific class of manifolds including $R^n$.

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on $r_M\left(x\right)$, the smallest eigenvalue of the Ricci tensor ${ric}_x$ in $x$, imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we construct the appropriate auxiliary functions to prove that the solution is monotone increasing in some direction in the unbounded domain. The different properties of the solution in bounded and unbounded domains are mainly attributed to the inherent non-locality of the fractional p-Laplacian.