Calculus of Variations and Partial Differential Equations最新文献

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}).

We establish some (C^{0,alpha }) and (C^{1,alpha }) regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane (Sigma ) as a power (a > -1) of the distance to (Sigma ). The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar (C^{1,alpha }) estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.

Let (w_0) be a bounded, (C^3), strictly plurisubharmonic function defined on (B_1subset mathbb {C}^n). Then (w_0) has a neighborhood in (L^{infty }(B_1)). Suppose that we have a function (phi ) in this neighborhood with (1-varepsilon le MA(phi )le 1+varepsilon ) and there exists a function u solving the linearized complex Monge–Amp(grave{text {e}})re equation: (det(phi _{kbar{l}})phi ^{ibar{j}}u_{ibar{j}}=0). Then there exist constants (alpha >0) and C such that (|u|_{C^{alpha }(B_{frac{1}{2}}(0))}le C), where (alpha >0) depends on n and C depends on n and (|u|_{L^{infty }(B_1(0))}), as long as (epsilon ) is small depending on n. This partially generalizes Caffarelli–Gutierrez’s estimate for linearized real Monge–Amp(grave{text {e}})re equation to the complex version.

In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 2004) proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using (L^1)-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino (Geom Topol 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp (L^p)-Sobolev and (L^p)-logarithmic Sobolev inequalities (both for (p>1) and (p=1)) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

In this paper, we prove interior Hessian estimates for shrinkers, expanders, translators, and rotators of the Lagrangian mean curvature flow under the assumption that the Lagrangian phase is hypercritical. We further extend our results to a broader class of Lagrangian mean curvature type equations.

In this paper we study the following (left( p,qright) )-Laplacian equation with critical exponent

where (1<qle p<r<p^{*}). After establishing ((PS)_c) condition for (cin (0,c^*)) for a certain constant (c^*) by employing the concentration compactness principle of Lions, multiple solutions for (lambda gg 1) are obtained by applying a critical point theorem due to Perera (J Anal Math, 2023. arxiv:2308.07901). A similar problem with subcritical exponents is also considered.

The mechanisms responsible for pattern formation have attracted a great deal of attention since Alan Turing elucidated his fascinating idea on diffusion-induced instability of steady states. Subsequent studies on the models demonstrated an entirely different class of solutions; namely localized structures composing of steadily moving fronts and pulses. In such energy-driven motion, the combination of short and long-range interaction plays an important ingredient for the generation of complex patterns. This competition on traveling wave dynamics, commonly observed in many physical and chemical phenomena, will be highlighted.