An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map (Brendle-Choi-Daskalopoulos, Acta Math. 219(2017): 1-16) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.
We introduce a level-set formulation for the mean curvature flow with obstacles and show existence and uniqueness of a viscosity solution. These results generalize a well known viscosity approach for the mean curvature flow without obstacle by Evans and Spruck and Chen, Giga and Goto in 1991. In addition, we show that this evolution is consistent with the variational scheme introduced by Almeida, Chambolle and Novaga (2012) and we study the long time behavior of our viscosity solutions.
A strong comparison principle for semicontinuous viscosity solutions of the prescribed mean curvature equation is considered. The difficulties of the problem come from the fact that this nonlinear equation is non-uniformly elliptic, does not depend on the value of unknown functions, depends on spatial variables and solutions are semicontinuous. Our simple proof of the strong comparison principle consists only of three ingredients, the definition of viscosity solutions, the inf and sup convolutions of functions, and the theory of classical solutions of quasilinear elliptic equations. Once we ha…
Here we provide a uniqueness result for viscosity solutions to sub-Riemannian mean curvature flow. In this setting the uniqueness cannot be deduced via comparison principle, which is known only for graphs and for radially symmetric surfaces, due to the presence of characteristic points. Here we prove that any viscosity solution is limit of a sequence of solutions of Riemannian flow, and obtain as a consequence uniqueness and comparison principle already known in the approximating riemannian setting. The results are provided in the settings of both 3-dimensional rototranslation group SE(2) and …
Here we provide uniqueness of vanishing viscosity solutions to sub-Riemannian mean curvature flow problem, which was known only far from characteristic points or under special symmetry condition. We employ vanishing viscosity approach and look for solutions as limit of solutions to approximating flow, which is well defined also at characteristic points, and estimate the rate of convergence of the approximating solutions. The results are provided in the settings of both 3-dimensional rototranslation group $\text{SE}(2)$ and Heisenberg group $\text{H}^1$ and they are particularly important due t…
A general purely crystalline mean curvature flow equation with a nonuniform driving force term is considered. The unique existence of a level set flow is established when the driving force term is continuous and spatially Lipschitz uniformly in time. By introducing a suitable notion of a solution a comparison principle of continuous solutions is established for equations including the level set equations. An existence of a solution is obtained by stability and approximation by smoother problems. A necessary equi-continuity of approximate solutions is established. It should be noted that the va…
By making use of the approximation method, we obtain the existence and regularity of the viscosity solutions for the generalized mean curvature flow. The asymptotic behavior of the flow is also considered. In particular, the Dirichlet problem of the degenerate elliptic equation $$ -|\nabla v|(\mathrm{div}(\frac{\nabla v}{|\nabla v|})+ν)=0 $$ is solvable in viscosity sense.
In this work we consider viscosity solutions to second order parabolic PDEs $u_{t}+F(t,x,u,du,d^{2}u)=0$ defined on compact Riemannian manifolds with boundary conditions. We prove comparison, uniqueness and existence results for the solutions. Under the assumption that the manifold $M$ has nonnegative sectional curvature, we get the finest results. If one additionally requires $F$ to depend on $d^{2}u$ in a uniformly continuous manner, the assumptions on curvature can be thrown away.
We consider nonlocal curvature functionals associated with positive interaction kernels, and we show that local anisotropic mean curvature functionals can be retrieved in a blow-up limit from them. As a consequence, we prove that the viscosity solutions to the rescaled nonlocal geometric flows locally uniformly converge to the viscosity solution to the anisotropic mean curvature motion. The result is achieved by combining a compactness argument and a set-theoretic approach related to the theory of De Giorgi's barriers for evolution equations.
In this paper we deal with singular varieties of bounded mean curvature in the viscosity sense. They contain all varifolds of bounded generalized mean curvature. In the first part we investigate the second-order properties of these varieties, obtaining results that are new also in the varifold's setting. In particular we prove that the generalized normal bundle of these varieties satisfies a natural Lusin (N) condition, which allows to extend the classical Coarea formula for the Gauss map of smooth varieties, and to introduce for all integral varifolds of bounded mean curvature a natural defin…
In this article we introduce a Partial Differential Equation (PDE) for the rank one convex envelope. Rank one convex envelopes arise in non-convex vector valued variational problems \cite{BallElasticity, kohn1986optimal1, BallJames87, chipot1988equilibrium}. More generally, we study a PDE for directional convex envelopes, which includes the usual convex envelope \cite{ObermanConvexEnvelope} and the rank one convex envelope as special cases. Existence and uniqueness of viscosity solutions to the PDE is established. Wide stencil elliptic finite difference schemes are built. Convergence of finite…
We show that every bounded subset of an Euclidean space can be approximated by a set that admits a certain vector field, the so-called Cahn-Hoffman vector field, that is subordinate to a given anisotropic metric and has a square-integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner. We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently in…
We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.
We show the existence of continuous viscosity solutions to the equation describing the flow of a graph in the Carnot group G x R according to its horizontal Gauss curvature. In doing so, we prove a comparison principle for degenerate parabolic equations of the form u_t + F(D_0u, (D_0^2u)^*) = 0 for u defined on G.
It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the first part of the paper, we show that locally convex solutions are exterior non-collapsing for a larger class of speed functions than previously considered; more precisely, we show that the previous results hold when convexity of the speed function is relaxed to inverse-concavity.…
In this note, we concentrate on the sub-Laplace on the nilpotent Lie group of rank two, which is the infinitesimal generator of the diffusion generated by $n$ Brownian motions and their $\frac{n(n-1)}2$ Lévy area processes, which is the simple extension of the sub-Laplace on the Heisenberg group $\mathbb{H}$. In order to study contraction properties of the heat kernel, we show that, as in the cases of the Heisenberg group and the three Brownian motion model, the restriction of the sub-Laplace acting on radial functions (see Definition \ref{radial fun}) satisfies a positive Ricci curvature cond…
We extend the theory of viscosity solutions to treat scalar-valued doubly-nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e., we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g., we prove the existence of a solution to a rate-independent level-s…
We address in this paper the study of a geometric evolution, corresponding to a curvature which is non-local and singular at the origin. The curvature represents the first variation of the energy recently proposed as a variant of the standard perimeter penalization for the denoising of nonsmooth curves. To deal with such degeneracies, we first give an abstract existence and uniqueness result for viscosity solutions of non-local degenerate Hamiltonians, satisfying suitable continuity assumption with respect to Kuratowsky convergence of the level sets. This abstract setting applies to an approxi…
The goal of this paper is manyfold. Firstly, we want to give a short introduction to the Bochner Laplacian and explain why it acts locally as a magnetic Laplacian. Secondly, given a confining magnetic field, we use Agmon-like estimates to reduce its spectral study to magnetic Laplacians, in the semiclassical limit. Finally, we use this to translate already-known spectral asymptotics for the magnetic Laplacian to the Bochner Laplacian.
We prove that viscosity solutions of geometric equations in step two Carnot groups can be equivalently reformulated by restricting the set of test functions at the singular points. These are characteristic points for the level sets of the solutions and are usually difficult to deal with. A similar property is known in the euclidian space, and in Carnot groups is based on appropriate properties of a suitable homogeneous norm. We also use this idea to extend to Carnot groups the definition of generalised flow, and it works similarly to the euclidian setting. These results simplify the handling o…
We study the following question: What is the largest deterministic amount of time $T_*$ that a suitably normalized martingale $X$ can be kept inside a convex body $K$ in $\mathbb{R}^d$? We show, in a viscosity framework, that $T_*$ equals the time it takes for the relative boundary of $K$ to reach $X(0)$ as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio--Soner codimen…
This paper aims at building a unified framework to deal with a wide class of local and nonlocal translation-invariant geometric flows. First, we introduce a class of generalized curvatures, and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows. We then introduce a class of generalized perimeters, whose first variation is an admissible generalized curvature. Within this class, we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set PDE. We also describe several examples…
