Calculus of Variations and Partial Differential Equations最新文献

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 ).

We study the elastic flow of closed curves and of open curves with clamped boundary conditions in the hyperbolic plane. While global existence and convergence toward critical points for initial data with sufficiently small energy is already known, this study pioneers an investigation into the flow’s singular behavior. We prove a convergence theorem without assuming smallness of the initial energy, coupled with a quantification of potential singularities: Each singularity carries an energy cost of at least 8. Moreover, the blow-ups of the singularities are explicitly classified. A further contribution is an explicit understanding of the singular limit of the elastic flow of (lambda )-figure-eights, a class of curves that previously served in showing sharpness of the energy threshold 16 for the smooth convergence of the elastic flow of closed curves.

We establish dispersive estimates and local decay estimates for the time evolution of non-self-adjoint matrix Schrödinger operators with threshold resonances in one space dimension. In particular, we show that the decay rates in the weighted setting are the same as in the regular case after subtracting a finite rank operator corresponding to the threshold resonances. Such matrix Schrödinger operators naturally arise from linearizing a focusing nonlinear Schrödinger equation around a solitary wave. It is known that the linearized operator for the 1D focusing cubic NLS equation exhibits a threshold resonance. We also include an observation of a favorable structure in the quadratic nonlinearity of the evolution equation for perturbations of solitary waves of the 1D focusing cubic NLS equation.

Our main result is a weighted fractional Poincaré–Sobolev inequality improving the celebrated estimate by Bourgain–Brezis–Mironescu. This also yields an improvement of the classical Meyers–Ziemer theorem in several ways. The proof is based on a fractional isoperimetric inequality and is new even in the non-weighted setting. We also extend the celebrated Poincaré–Sobolev estimate with (A_p) weights of Fabes–Kenig–Serapioni by means of a fractional type result in the spirit of Bourgain–Brezis–Mironescu. Examples are given to show that the corresponding (L^p)-versions of weighted Poincaré inequalities do not hold for (p>1).

In this paper, we introduce and analyse an explicit formulation of fractional powers of the parabolic Lamé operator and we then study the extension problem associated to such non-local operators. We also study the various regularity properties of solutions to such an extension problem via a transformation as in Ang et al. (Commun Partial Differ Equ 23:371–385, 1998), Alessandrini and Morassi (Commun Partial Differ Equ 26(9–10):1787–1810, 2001), Eller et al. (Nonlinear partial differential equations andtheir applications, North-Holland, Amsterdam, 2002), and Gurtin (in: Truesdell, C. (ed.) Handbuch der Physik, Springer, Berlin, 1972), which reduces the extension problem for the parabolic Lamé operator to another system that resembles the extension problem for the fractional heat operator. Finally in the case when (s ge 1/2), by proving a conditional doubling property for solutions to the corresponding reduced system followed by a blowup argument, we establish a space-like strong unique continuation result for (mathbb {H}^s textbf{u}=Vtextbf{u}).