Second order rectifiability of varifolds of bounded mean curvature

We prove that the support of an m dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $$ {\mathscr {H}}^{m} $$ H m almost everywhere by a countable union of m dimensional submanifolds of class $$ {\mathcal {C}}^{2} $$ C 2 . The $$ {\mathcal {C}}^{2} $$ C 2 -regularity of the submanifolds is optimal.

Fujita type results for quasilinear parabolic inequalities with nonlocal terms

<p style='text-indent:20px;'>In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} &amp;u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\ &amp;u(x,0) = u_0(x)\ge0 \,\, \text{ in } \mathbb R^N,\end{cases} \qquad (P^{\pm}) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ u_0\in L^1_{loc}({\mathbb R}^N) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ L_{\mathcal{A}} $\end{document}</tex-math></inline-formula> denotes a weakly <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>-coercive operator, which includes as prototype the <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>-Laplacian or the generalized mean curvature operator, <inline-formula><tex-math id="M5">\begin{document}$ p,\,q&gt;0 $\end{document}</tex-math></inline-formula>, while <inline-formula><tex-math id="M6">\begin{document}$ K\ast u^p $\end{document}</tex-math></inline-formula> stands for the standard convolution operator between a weight <inline-formula><tex-math id="M7">\begin{document}$ K&gt;0 $\end{document}</tex-math></inline-formula> satisfying suitable conditions at infinity and <inline-formula><tex-math id="M8">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>. For problem <inline-formula><tex-math id="M9">\begin{document}$ (P^-) $\end{document}</tex-math></inline-formula> we obtain a Fujita type exponent while for <inline-formula><tex-math id="M10">\begin{document}$ (P^+) $\end{document}</tex-math></inline-formula> we show that no such critical exponent exists. Our approach relies on nonlinear capacity estimates adapted to the nonlocal setting of our problems. No comparison results or maximum principles are required.</p>

Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

A diffused interface with the advection term in a Sobolev space

We study the asymptotic limit of diffused surface energy in the van der Waals–Cahn–Hillard theory when an advection term is added and the energy is uniformly bounded. We prove that the limit interface is an integral varifold and the generalized mean curvature vector is determined by the advection term. As the application, a prescribed mean curvature problem is solved using the min–max method.

Generalized Lagrangian mean curvature flows: The cotangent bundle case

In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel {(n,0)} -form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

A Class of Parabolic Equations Driven by the Mean Curvature Flow

We study a class of parabolic equations which can be viewed as a generalized mean curvature flow acting on cylindrically symmetric surfaces with a Dirichlet condition on the boundary. We prove the existence of a unique solution by means of an approximation scheme. We also develop the theory of asymptotic stability for solutions of general parabolic problems.

Nonhomogeneous Nonlinear Dirichlet Problems with ap-Superlinear Reaction

We consider a nonlinear Dirichlet elliptic equation driven by a nonhomogeneous differential operator and with a Carathéodory reactionf(z,ζ), whose primitivef(z,ζ)isp-superlinear near±∞, but need not satisfy the usual in such cases, the Ambrosetti-Rabinowitz condition. Using a combination of variational methods with the Morse theory (critical groups), we show that the problem has at least three nontrivial smooth solutions. Our result unifies the study of “superlinear” equations monitored by some differential operators of interest like thep-Laplacian, the(p,q)-Laplacian, and thep-generalized mean curvature operator.

NONLINEAR WEIGHTED p-LAPLACIAN ELLIPTIC INEQUALITIES WITH GRADIENT TERMS

In this paper, we give sufficient conditions for the existence and nonexistence of nonnegative nontrivial entire weak solutions of p-Laplacian elliptic inequalities, with possibly singular weights and gradient terms, of the form div {g(|x|)|Du|p-2Du} ≥ h(|x|)f(u)ℓ(|Du|). We achieve our conclusions by using a generalized version of the well-known Keller–Ossermann condition, first introduced in [2] for the generalized mean curvature case, and in [11, Sec. 4] for the nonweighted p-Laplacian equation. Several existence results are also proved in Secs. 2 and 3, from which we deduce simple criteria of independent interest stated in the Introduction.