Calculus of Variations and Partial Differential Equations最新文献

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.

Whether the global existence and uniqueness of strong solutions to n-dimensional incompressible magnetohydrodynamic (MHD for short) equations with only kinematic viscosity or magnetic diffusion holds true or not remains an outstanding open problem. In recent years, stared from the pioneer work by Lin and Zhang (Commun Pure Appl Math 67(4):531–580, 2014), much more attention has been paid to the case when the magnetic field close to an equilibrium state (the background magnetic field for short). Specifically, when the background magnetic field satisfies the Diophantine condition (see (1.2) for details), Chen et al. (Sci China Math 41:1–10, 2022) first studied the perturbation system and established the decay estimates and asymptotic stability of its solutions in 3D periodic domain (mathbb {T}^3), which was then improved to (H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2)) for 2D periodic domain (mathbb {T}^2) and any (alpha >0), (beta >0) by Zhai (J Differ Equ 374:267–278, 2023). In this paper, we seek to find the optimal decay estimates and improve the space where the global stability is taking place. Through deeply exploring and effectively utilizing the structure of perturbation system, we discover a new dissipative mechanism, which enables us to establish the decay estimates in the Sobolev spaces with much lower regularity. Based on the above discovery, we greatly reduce the initial regularity requirement of aforesaid two works from (H^{4r+7}(mathbb {T}^3)) and (H^{(3+2beta )r+5+(alpha +2beta )}(mathbb {T}^2)) to (H^{(3r+3)^+}(mathbb {T}^n)) for (r>n-1) when (n=3) and (n=2) respectively. Additionally, we first present the linear stability result via the method of spectral analysis in this paper. From which, the decay estimates obtained for the nonlinear system can be seen as sharp in the sense that they are in line with those for the linearized system.

Variational models for cohesive fracture are based on the idea that the fracture energy is released gradually as the crack opening grows. Recently, [21] proposed a variational approximation via (Gamma )-convergence of a class of cohesive fracture energies by phase-field energies of Ambrosio-Tortorelli type, which may be also used as regularization for numerical simulations. In this paper we address the question of the asymptotic behaviour of critical points of the phase-field energies in the one-dimensional setting: we show that they converge to a selected class of critical points of the limit functional. Conversely, each critical point in this class can be approximated by a family of critical points of the phase-field functionals.