CVGMT Papershttp://cvgmt.sns.it/papers/en-usWed, 21 Feb 2018 22:45:26 +0000- A gradient flow approach to relaxation rates for the multi-dimensional Cahn-Hilliard equationhttp://cvgmt.sns.it/paper/3778/L. De Luca, M. Goldman, M. Strani.<p>The aim of this paper is to study relaxation rates for the Cahn-Hilliard equation in dimension larger than one. We follow the approach of Otto and Westdickenberg based on the gradient flow structure of the equation and establish differential and algebraic relationships between the energy,the dissipation, and the squared $\dot H^{-1}$ distance to a kink. This leads to a scale separation of the dynamics into two different stages: a first {\it fast} phase of the order $t^{-\frac 1 2}$ where one sees convergence to some kink, followed by a {\it slow} relaxation phase with rate $t^{-\frac 1 4}$ where convergence to the centered kink is observed.</p>http://cvgmt.sns.it/paper/3778/
- Boundary regularity of mass-minimizing integral currents and a question of Almgrenhttp://cvgmt.sns.it/paper/3777/C. De Lellis, G. De Philippis, J. Hirsch, A. Massaccesi.<p>This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ($C^{3, a_0}$ for a positive $a_0$ suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general weaker that the interior regularity.Moreover we remark an interesting elementary byproduct on boundary monotonicity formulae.</p>http://cvgmt.sns.it/paper/3777/
- Self-contracted curves are gradient flows of convex functionshttp://cvgmt.sns.it/paper/3776/E. Durand-Cartagena, A. Lemenant.<p>In this paper we prove that any $C^{1,\alpha}$ curve in $\mathbb{R^n}$, with $\alpha \in \left(\frac{1}{2},1\right]$, is the solution of the gradient flow equation for some $C^1$ convex function $f$, if and only if it is strongly self-contracted.</p>http://cvgmt.sns.it/paper/3776/
- Weakly localized states for nonlinear Dirac equationshttp://cvgmt.sns.it/paper/3775/W. Borrelli.<p>We prove the existence of infinitely many non square-integrable stationary solutions for a family of massless Dirac equations in 2D. They appear as effective equations in two dimensional honeycomb structures. We give a direct existence proof thanks to a particular radial ansatz, which also allows to provide the exact asymptotic behavior of spinor components. Moreover, those solutions admit a variational characterization. We also indicate how the content of the present paper allows to extend our previous results for the massive case to more general nonlinearities.</p>http://cvgmt.sns.it/paper/3775/
- Borderline gradient continuity of minimahttp://cvgmt.sns.it/paper/3774/P. Baroni, T. Kuusi, G. Mingione.<p>The gradient of any local minimiser of functionals of the type\[w \mapsto \int_\Omega f(x,w,Dw)\,dx+\int_\Omega w\mu\,dx,\]where $f$ has $p$-growth, $p>1$, and $\Omega \subset \mathbb R^n$, is continuous provided the optimal Lorentz space condition $\mu \in L(n,1)$ is satisfied and $x\to f(x, \cdot)$ is suitably Dini-continuous.</p>http://cvgmt.sns.it/paper/3774/
- Regularity for general functionals with double phasehttp://cvgmt.sns.it/paper/3773/P. Baroni, M. Colombo, G. Mingione.<p>We prove sharp regularity results for a general class of functionals of the type \[w \mapsto \int F(x, w, Dw) \, dx\;,\]featuring non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral \[w \mapsto \int b(x,w)(<br>Dw<br>^p+a(x)<br>Dw<br>^q) \, dx\;,\quad 1 <p < q\,, \quad a(x)\geq 0\;,\]with $0<\nu \leq b(\cdot)\leq L $. This changes its ellipticity rate according to the geometry of the level set $\{a(x)=0\}$ of the modulating coefficient $a(\cdot)$. We also present new methods and proofs, that are suitable to build regularity theorems for larger classes of non-autonomous functionals. Finally, we disclose some new interpolation type effects that, as we conjecture, should draw a general phenomenon in the setting of non-uniformly elliptic problems. Such effects naturally connect with the Lavrentiev phenomenon.</p>http://cvgmt.sns.it/paper/3773/
- Non-autonomous functionals, borderline cases and related function classeshttp://cvgmt.sns.it/paper/3772/P. Baroni, M. Colombo, G. Mingione.<p>We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with $(p,q)$-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima.</p>http://cvgmt.sns.it/paper/3772/
- Harnack inequalities for double phase functionalshttp://cvgmt.sns.it/paper/3771/P. Baroni, M. Colombo, G. Mingione.<p>We prove a Harnack inequality for minimizers of a class of non- autonomous functionals with non-standard growth conditions. They are characterized by the fact that their energy density switches between two types of different degenerate phases.</p>http://cvgmt.sns.it/paper/3771/
- Flatness results for nonlocal phase transitionshttp://cvgmt.sns.it/paper/3770/E. Cinti.<p>We consider a nonlocal version of the Allen-Cahn equation, which models phase transitions problems. In the classical setting, the connection between the Allen-Cahn equation and the classification of entire minimal surfaces is well known and motivates a celebrated conjecture by E. De Giorgi on the one-dimensional symmetry of bounded monotone solutions to the (classical) Allen-Cahn equation up to dimension 8. In this note, we present some recent results in the study of the nonlocal analogue of this phase transition problem. In particular we describe the results obtained in several contributions <a href='8, 9, 13, 14, 25, 41, 44, 46'>8, 9, 13, 14, 25, 41, 44, 46</a> where the classification of certain entire bounded solutions to the fractional Allen-Cahn equation has been obtained. Moreover we describe the connection between the fractional Allen-Cahn equation and the fractional perimeter functional, and we present also some results in the classifications of nonlocal minimal surfaces obtained in <a href='16, 42, 10, 21'>16, 42, 10, 21</a>.</p>http://cvgmt.sns.it/paper/3770/
- Regularity of Lagrangian flows over $RCD^*(K,N)$ spaceshttp://cvgmt.sns.it/paper/3769/E. BruĂ¨, D. Semola.<p>The aim of this note is to provide regularity results for Regular Lagrangian flows of Sobolevvector fields over compact metric measure spaces verifying the Riemannian curvature dimensioncondition.We first prove, borrowing some ideas already present in the literature, that flows generated byvector fields with bounded symmetric derivative are Lipschitz, providing the natural extensionof the standard Cauchy-Lipschitz theorem to this setting. Then we prove a Lusin-type regularityresult in the Sobolev case (under the additional assumption that the m.m.s. is Ahlfors regular)therefore extending the already known Euclidean result.</p>http://cvgmt.sns.it/paper/3769/
- Stable solutions to some elliptic problems: minimal cones, the Allen-Cahn equation, and blow-up solutionshttp://cvgmt.sns.it/paper/3768/X. CabrĂ©, G. Poggesi.<p> These notes record the lectures for the CIME Summer Course taught by thefirst author in Cetraro during the week of June 19-23, 2017. The notes containthe proofs of several results on the classification of stable solutions to somenonlinear elliptic equations. The results are crucial steps within theregularity theory of minimizers to such problems. We focus our attention onthree different equations, emphasizing that the techniques and ideas in thethree settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality ofthe Simons cone in high dimensions, and we give almost all details in the proofof J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed byE. De Giorgi in 1978. This is our second problem, for which we discuss someresults, as well as an open problem in high dimensions on the saddle-shapedsolution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns theboundedness of stable solutions to reaction-diffusion equations in boundeddomains. We present proofs on their regularity in low dimensions and discussthe main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, freeboundary problems, and nonlocal minimal surfaces.</p>http://cvgmt.sns.it/paper/3768/
- Compactness and lower semicontinuity in $GSBD$http://cvgmt.sns.it/paper/3767/A. Chambolle, V. Crismale.<p>In this paper, we prove a compactness and semicontinuityresult in $GSBD$ for sequences with bounded Griffith energy.This generalises classical results in $(G)SBV$ by Ambrosio and $SBD$ by Bellettini-Coscia-Dal Maso.As a result, the static problem in Francfort-Marigo's variational approach to crack growth admits (weak) solutions.Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies, for which we have recently shown the $\Gamma$-convergence to Griffith energy.</p>http://cvgmt.sns.it/paper/3767/
- Ground states and concentration of mass in stationary Mean Field Games with superlinear Hamiltonianshttp://cvgmt.sns.it/paper/3766/A. Cesaroni, M. Cirant.<p>In this paper we provide the existence of classical solutions to stationary mean field game systems in the whole space $\mathbb{R}^N$, with coercive potential, aggregating local coupling, and under general conditions on the Hamiltonian, completing the analysis started in the companion paper <a href='http://cvgmt.sns.it/paper/3469/'>6</a>.The only structural assumption we make is on the growth at infinity of the coupling term in terms of the growth of the Hamiltonian.This result is obtained using a variational approach based on the analysis of the non-convex energy associated to the system. Finally, we show that in the vanishing viscosity limit mass concentrates around the flattest minima of the potential, and that the asymptotic shape of the solutions in a suitable rescaled setting converges to a ground state, i.e. a classical solution to a mean field game system without potential.</p>http://cvgmt.sns.it/paper/3766/
- Loss of regularity for the continuity equation with non-Lipschitz velocity fieldhttp://cvgmt.sns.it/paper/3765/G. Alberti, G. Crippa, A. L. Mazzucato.<p>We consider the Cauchy problem for the continuity equation in space dimension ${d \geq 2}$. We construct a divergence-free velocity field uniformly bounded in all Sobolev spaces $W^{1,p}$, for $1 \leq p<\infty$, and a smooth compactly supported initial datumsuch that the unique solution to the continuity equation with this initial datum and advecting field does not belong to any Sobolev space of positive fractional order at any positive time.We also construct velocity fields in $W^{r,p}$, with $r>1$, and solutions of the continuity equation with these velocities that exhibit some loss of regularity, as long as the Sobolev space $W^{r,p}$ does not embed in the space of Lipschitz functions.Our constructions are based on examples of optimal mixers from the companion paper "Exponential self-similar mixing by incompressible flows" (Preprint arXiv:1605.02090), and have been announced in "Exponential self-similar mixing and loss of regularity for continuity equations" (C. R. Math. Acad. Sci. Paris, 352 (2014), no. 11).</p>http://cvgmt.sns.it/paper/3765/
- Second order differentiation formula on $RCD^*(K,N)$ spaceshttp://cvgmt.sns.it/paper/3764/N. Gigli, L. Tamanini.<p>Aim of this paper is to prove the second order differentiation formula for $H^{2,2}$ functions along geodesics in $RCD^*(K,N)$ spaces with $N < \infty$. This formula is new even in the context of Alexandrov spaces, where second order differentiation is typically related to semiconvexity.</p><p>We establish this result by showing that $W_2$-geodesics can be approximated up to second order, in a sense which we shall make precise, by entropic interpolation. In turn this is achieved by proving new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:</p><p>- equiboundedness of the densities along the entropic interpolations,</p><p>- local equi-Lipschitz continuity of the Schr\"odinger potentials,</p><p>- a uniform weighted $L^2$ control of the Hessian of such potentials.</p><p>Finally, the techniques adopted in this paper can be used to show that in the $RCD$ setting the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case.</p><p>With respect to a previous version, where the space was assumed to be compact, in this paper the second order differentiation formula is proved in full generality.</p>http://cvgmt.sns.it/paper/3764/
- On critical points of the relative fractional perimeterhttp://cvgmt.sns.it/paper/3762/A. Malchiodi, M. Novaga, D. Pagliardini.<p>Abstract. We study the localization of sets with constant nonlocal mean curvatureand prescribed small volume in a bounded open set with smooth boundary, proving thatthey are sufficiently close to critical points of a suitable non-local potential. We thenconsider the fractional perimeter in half-spaces. We prove the existence of a minimizerunder fixed volume constraint, showing some of its properties such as smoothness andsymmetry, being a graph in the $x_N$-direction, and characterizing its intersection withthe hyperplane $\{x_N = 0\}$.</p>http://cvgmt.sns.it/paper/3762/
- (Log-)epiperimetric inequality and regularity over smooth cones for almost Area-Minimizing currentshttp://cvgmt.sns.it/paper/3761/M. Engelstein, L. Spolaor, B. Velichkov.<p>In this paper we prove a new logarithmic epiperimetric inequality for multiplicity-one stationary cones with isolated singularity by flowing radially any nearby trace along appropriately chosen directions in the sphere. In contrast to previous epiperimetric inequalities for minimal surfaces by Reifenberg (Ann. of Math. 1964), Taylor (Invent. Math. 1973, Ann. of Math. 1976) and White (Duke Math. J. 1983), we need no a priori assumptions on the structure of the cone (e.g. integrability). If the cone is integrable (not only through rotations), we recover the classical epiperimetric inequality. As a consequence we deduce a new epsilon-regularity result for almost area-minimizing currents at singular points where at least one blow-up is a multiplicity-one cone with isolated singularity. This result is similar to the one for stationary varifolds of L. Simon (Ann. of Math. 1983), but independent from it since almost minimizers do not satisfy any equation.</p>http://cvgmt.sns.it/paper/3761/
- Optimal Besov differentiability for entropy solutions of the eikeonal equationhttp://cvgmt.sns.it/paper/3760/F. Ghiraldin, X. Lamy.<p>In this paper we study the Eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production and the validity of a kinetic formulation.</p>http://cvgmt.sns.it/paper/3760/
- Damage model for plastic materials at finite strainshttp://cvgmt.sns.it/paper/3759/D. Melching, R. Scala, J. Zeman.<p>We consider a model for nonlinear elastoplasticity coupled with incomplete damage. The internal energy of the deformed elastoplastic body depends on the deformation $y$, on the plastic strain $P$, and on an internal variable $z$ describing the damage level of the medium. We consider a dissipation distance $D$ between internal states accounting for coupled plastic deformation and damage. Moving from time-discretization we prove the existence of a rate-independent quasistatic evolution of the system.</p>http://cvgmt.sns.it/paper/3759/
- Heat content in non-compact Riemannian manifoldshttp://cvgmt.sns.it/paper/3758/M. van den Berg.http://cvgmt.sns.it/paper/3758/