Calculus of Variations and Partial Differential Equations最新文献

In this paper, we study the rigidity problem for compact minimal Legendrian submanifolds in the unit Euclidean spheres via eigenvalues of fundamental matrices, which measure the squared norms of the second fundamental form on all normal directions. By using Lu’s inequality (Lu in J Funct Anal 261:1284–1308, 2011) on the upper bound of the squared norm of Lie brackets of symmetric matrices, we establish an optimal pinching theorem for such submanifolds of all dimensions, giving a new characterization for the Calabi tori. This pinching condition can also be described by the eigenvalues of the Ricci curvature tensor. Moreover, when the third large eigenvalue of the fundamental matrix vanishes everywhere, we get an optimal rigidity theorem under a weaker pinching condition.

We prove a new existence theorem for proper solutions of Huisken and Ilmanen’s weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.

The (L^{p}) dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the (L^{p}) dual Minkowski problem, asks about existence of a convex body with prescribed (L^{p}) dual curvature measure. This question unifies the previously disjoint (L^{p}) Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the (L^{p}) dual Minkowski problem for the case of (q<p+1), (-1<p<0), and (pne q) for even measures.

We study a singular limit of the classical parabolic-elliptic Patlak-Keller-Segel (PKS) model for chemotaxis with non linear diffusion. The main result is the (Gamma ) convergence of the corresponding energy functional toward the perimeter functional. Following recent work on this topic, we then prove that under an energy convergence assumption, the solution of the PKS model converges to a solution of the Hele-Shaw free boundary problem with surface tension, which describes the evolution of the interface separating regions with high density from those with low density. This result complements a recent work by the author with I. Kim and Y. Wu, in which the same free boundary problem is derived from the congested PKS model (which includes a density constraint (rho le 1) and a pressure term): It shows that the congestion constraint is not necessary to observe phase separation and surface tension phenomena.

This work is concerned with the nonexistence of nontrivial nonnegative weak solutions for a strongly p-coercive elliptic differential inequality with weighted nonlocal source and gradient absorption terms in the whole space. Under the condition that the positive weight in the absorption term is either a sufficiently small constant or more general, we establish new Liouville type results containing the critical case. The key ingredient in the proof is the rescaled test function method developed by Mitidieri and Pohozaev.

We generalize the results of Kuwae–Shioya and Bačák on Mosco convergence established for CAT(0)-spaces to the CAT(1)-setting, so that Mosco convergence implies convergence of resolvents which in turn imply convergence of gradient flows for lower-semicontinuous semi-convex functions. Our techniques utilize weak convergence in CAT(1)-spaces and also cover asymptotic relations of sequences of such spaces introduced by Kuwae-Shioya, including Gromov–Hausdorff limits.

In this paper, we prove the isoperimetric inequality for the anisotropic Gaussian measure and characterize the cases of equality. We also find an example that shows Ehrhard symmetrization fails to decrease for the anisotropic Gaussian perimeter and gives a new inequality that includes an error term. This new inequality, in particular, gives us a hint to prove a uniqueness result for the anisotropic Ehrhard symmetrization.

In this paper, we deal with the following class of ((p_{1}, p_{2}))-Laplacian problems:

where (Nge 2), (1<p_{1}<p_{2}le N), (Delta _{p_{i}}) is the (p_{i})-Laplacian operator, for (i=1, 2), and (g:mathbb {R}rightarrow mathbb {R}) is a Berestycki-Lions type nonlinearity. Using appropriate variational arguments, we obtain the existence of a ground state solution. In particular, we provide three different approaches to deduce this result. Finally, we prove the existence of infinitely many radially symmetric solutions. Our results improve and complement those that have appeared in the literature for this class of problems. Furthermore, the arguments performed throughout the paper are rather flexible and can be also applied to study other p-Laplacian and ((p_1, p_2))-Laplacian equations with general nonlinearities.

We establish the Hölder estimate and the asymptotic behavior at infinity for K-quasiconformal mappings over exterior domains in (mathbb {R}^2). As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in (mathbb {R}^2).

We study ground states of a relativistic Fermi system involved with the pseudo-differential operator (sqrt{-c^2Delta +c^4m^2}-c^2m) in the (L^2)-subcritical case, where (m>0) denotes the rest mass of fermions, and (cge 1) represents the speed of light. By employing Green’s function and the variational principle of many-fermion systems, we prove the existence of ground states for the system. The asymptotic behavior of ground states for the system is also analyzed in the non-relativistic limit where (crightarrow infty ).