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Abstract: We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals. PubDate: 2022-01-16

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Abstract: As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we study the equivariant Yamabe problem with boundary. PubDate: 2022-01-16

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Abstract: We give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver. PubDate: 2022-01-16

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Abstract: The paper studies a dynamic blocking problem, motivated by a model of optimal fire confinement. While the fire can expand with unit speed in all directions, barriers are constructed in real time. An optimal strategy is sought, minimizing the total value of the burned region, plus a construction cost. It is well known that optimal barriers exists. In general, they are a countable union of compact, connected, rectifiable sets. The main result of the present paper shows that optimal barriers are nowhere dense. The proof relies on new estimates on the reachable sets and on optimal trajectories for the fire, solving a minimum time problem in the presence of obstacles. PubDate: 2022-01-15

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Abstract: In this paper, we present first insights about the Dirichlet-to-Neumann operator in \(L^{1}\) associated with the 1-Laplace operator or total variational flow operator. This operator is the main object, for example, in studying inverse problems related to image processing, but also admits an important relation to geometry. We show that this operator can be represented by the sub-differential in \(L^1\times L^{\infty }\) of a convex, homogeneous, and continuous functional on \(L^{1}\) . This is quite surprising since it implies a type of stability or compactness result even though the singular Dirichlet problem governed by the 1-Laplace operator might have infinitely many weak solutions (if the given boundary data is not continuous). As an application, we obtain well-posedness and long-time stability of solutions of a singular coupled elliptic-parabolic initial boundary-value problem. PubDate: 2022-01-15

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Abstract: In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the stack of cubes covering argument which is tailored for the nonlinearity that we consider. PubDate: 2022-01-04

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Abstract: We study the minimum problem for functionals of the form $$\begin{aligned} \mathcal {F}(u) = \int _{I} f(x, u(x), u^\prime (x))\,dx, \end{aligned}$$ where the integrand \(f:I\times \mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}\) is not convex in the last variable. We provide an existence result assuming that the lower convex envelope \(\overline{f}=\overline{f}(x,p,\xi )\) of f satisfies a suitable affinity condition on the set on which \(f>\overline{f}\) and that the map \(p_i\mapsto f(x,p,\xi )\) is monotone with respect to one single component \(p_i\) of the vector p. We show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the existence of minimizers. PubDate: 2022-01-04

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Abstract: In this article we study the homogenization rates of eigenvalues of a Steklov problem with rapidly oscillating periodic weight functions. The results are obtained via a careful study of oscillating functions on the boundary and a precise estimate of the \(L^\infty \) bound of eigenfunctions. As an application we provide some estimates on the first nontrivial curve of the Dancer–Fučík spectrum. PubDate: 2022-01-04

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Abstract: We consider the elliptic system problems arising from the Maxwell-Chern-Simons model. In this system, there are two important parameters related to Chern-Simons mass scale and electric charge. Under almost optimal conditions on these two parameters, we show the existence of nontopological condensates with magnetic field concentrated at multi-bubbling points. PubDate: 2022-01-04

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Abstract: The Minkowski problem in Gaussian probability space studied by Huang, Xi, and Zhao is generalized to the \(L_p\) -Gaussian Minkowski problem in this paper. The existence and uniqueness of o-symmetric weak solution in the case \(p\ge 1\) is obtained. PubDate: 2022-01-04

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Abstract: We provide a new convergence proof of the celebrated Merriman–Bence–Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoḡlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation. PubDate: 2022-01-04

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Abstract: We consider Newton’s problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton’s problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton’s problem, and we show that they are not. PubDate: 2022-01-04

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Abstract: We study degenerate fully nonlinear free transmission problems, where the degeneracy rate varies in the domain. We prove optimal pointwise regularity depending on the degeneracy rate. Our arguments consist of perturbation methods, relating our problem to a homogeneous, fully nonlinear, uniformly elliptic equation. PubDate: 2022-01-04

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Abstract: We consider weak solutions with finite entropy production to the scalar conservation law $$\begin{aligned} \partial _t u+\mathrm {div}_x F(u)=0 \quad \text{ in } (0,T)\times \mathbb {R}^d. \end{aligned}$$ Building on the kinetic formulation we prove under suitable nonlinearity assumption on f that the set of non Lebesgue points of u has Hausdorff dimension at most d. A notion of Lagrangian representation for this class of solutions is introduced and this allows for a new interpretation of the entropy dissipation measure. PubDate: 2022-01-04

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Abstract: We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure. PubDate: 2022-01-04

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Abstract: We present some geometric applications, of global character, of the bubbling analysis developed by Buzano and Sharp for closed minimal surfaces, obtaining smooth multiplicity one convergence results under upper bounds on the Morse index and suitable lower bounds on either the genus or the area. For instance, we show that given any Riemannian metric of positive scalar curvature on the three-dimensional sphere the class of embedded minimal surfaces of index one and genus \(\gamma \) is sequentially compact for any \(\gamma \ge 1\) . Furthemore, we give a quantitative description of how the genus drops as a sequence of minimal surfaces converges smoothly, with mutiplicity \(m\ge 1\) , away from finitely many points where curvature concentration may happen. This result exploits a sharp estimate on the multiplicity of convergence in terms of the number of ends of the bubbles that appear in the process. PubDate: 2021-12-10 DOI: 10.1007/s00526-021-02135-x

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Abstract: We consider linear partial differential equations on resistance spaces that are uniformly elliptic and parabolic in the sense of quadratic forms and involve abstract gradient and divergence terms. Our main interest is to provide graph and metric graph approximations for their unique solutions. For families of equations with different coefficients on a single compact resistance space we prove that solutions have accumulation points with respect to the uniform convergence in space, provided that the coefficients remain bounded. If in a sequence of equations the coefficients converge suitably, the solutions converge uniformly along a subsequence. For the special case of local resistance forms on finitely ramified sets we also consider sequences of resistance spaces approximating the finitely ramified set from within. Under suitable assumptions on the coefficients (extensions of) linearizations of the solutions of equations on the approximating spaces accumulate or even converge uniformly along a subsequence to the solution of the target equation on the finitely ramified set. The results cover discrete and metric graph approximations, and both are discussed. PubDate: 2021-12-10 DOI: 10.1007/s00526-021-02119-x

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Abstract: We study local minima of the p-conformal energy functionals, $$\begin{aligned} {\mathsf {E}}_{{{\mathcal {A}}}}^*(h){:}{=}\int _{\mathbb D}{{{\mathcal {A}}}}({\mathbb K}(w,h)) \;J(w,h) \; dw,\quad h _{\mathbb S}=h_0 _{\mathbb S}, \end{aligned}$$ defined for self mappings \(h:{\mathbb D}\rightarrow {\mathbb D}\) with finite distortion of the unit disk with prescribed boundary values \(h_0\) . Here \({\mathbb K}(w,h) = \frac{\Vert Dh(w)\Vert ^2}{J(w,h)} \) is the pointwise distortion functional, and \({{{\mathcal {A}}}}:[1,\infty )\rightarrow [1,\infty )\) is convex and increasing with \({{{\mathcal {A}}}}(t)\approx t^p\) for some \(p\ge 1\) , with additional minor technical conditions. Note \({{{\mathcal {A}}}}(t)=t\) is the Dirichlet energy functional. Critical points of \({\mathsf {E}}_{{{\mathcal {A}}}}^*\) satisfy the Ahlfors-Hopf inner-variational equation $$\begin{aligned} {{{\mathcal {A}}}}'({\mathbb K}(w,h)) h_w \overline{h_{{\overline{w}}}} = \Phi \end{aligned}$$ where \(\Phi \) is a holomorphic function. Iwaniec, Kovalev and Onninen established the Lipschitz regularity of critical points. Here we give a sufficient condition to ensure that a local minimum is a diffeomorphic solution to this equation, and that it is unique. This condition is necessarily satisfied by any locally quasiconformal critical point, and is basically the assumption \({\mathbb K}(w,h)\in L^1({\mathbb D})\cap L^r_{loc}({\mathbb D})\) for some \(r>1\) . PubDate: 2021-12-10 DOI: 10.1007/s00526-021-02121-3

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Abstract: We establish the Krylov–Safonov Harnack inequalities and Hölder estimates for fully nonlinear nonlocal operators of non-divergence form on Riemannian manifolds with nonnegative sectional curvatures. To this end, we first define the nonlocal Pucci operators on manifolds that give rise to the concept of non-divergence form operators. We then provide the uniform regularity estimates for these operators which recover the classical estimates for second order local operators as limits. PubDate: 2021-12-10 DOI: 10.1007/s00526-021-02124-0

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Abstract: Let \(\{M_k\}_{k=1}^{\infty }\) be a sequence of closed, singular, minimal hypersurfaces in a closed Riemannian manifold \((N^{n+1},g), n+1 \ge 3\) . Suppose, the volumes of \(M_k\) are uniformly bounded from above and the pth Jacobi eigenvalues \(\lambda _p\) ’s of \(M_k\) are uniformly bounded from below. Then we will prove that there exists a closed, singular, minimal hypersurface M in N, with the above-mentioned volume and eigenvalue bounds, such that, possibly after passing to a subsequence, \(M_k\) weakly converges (in the sense of varifolds) to M, possibly with multiplicities. Moreover, the convergence is smooth and graphical over the compact subsets of \(reg(M) \setminus {\mathcal {Y}}\) , where \({\mathcal {Y}}\) is a finite subset of reg(M) with \( {\mathcal {Y}} \le p-1\) . As a corollary, we get the compactness of the space of closed, singular, minimal hypersurfaces with uniformly bounded volume and index. These results generalize the previous theorems of Ambrozio–Carlotto–Sharp (J Geom Anal 26(4):2591–2601, 2016) and Sharp (J Differ Geom 106(2):317–339, 2017) in higher dimensions. We will also show that if \(\varSigma \) is a singular, minimal hypersurface with \({\mathcal {H}}^{n-2}(sing(\varSigma ))=0\) , then the index of the varifold associated to \(\varSigma \) coincides with the index of \(reg(\varSigma )\) (with respect to compactly supported normal vector fields on \(reg(\varSigma )\) ). PubDate: 2021-12-10 DOI: 10.1007/s00526-021-02136-w