cvgmt Papershttp://cvgmt.sns.it/papers/en-usSun, 18 Nov 2018 20:22:19 +0000Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentationhttp://cvgmt.sns.it/paper/4117/P. R. A. S. A. N. T. A. K. U. M. A. R. BARIK.<p> In this paper we study the continuous coagulation and multiple fragmentationequation for the mean-field description of a system of particles taking intoaccount the combined effect of the coagulation and the fragmentation processesin which a system of particles growing by successive mergers to form a biggerone and a larger particle splits into a finite number of smaller pieces. Wedemonstrate the global existence of mass-conserving weak solutions for a wideclass of coagulation rate, selection rate and breakage function. Here, both thebreakage function and the coagulation rate may have algebraic singularity onboth the coordinate axes. The proof of the existence result is based on a weakL<sup>1</sup> compactness method for two different suitable approximations to theoriginal problem, i.e. the conservative and non-conservative approximations.Moreover, the mass-conservation property of solutions is established for bothapproximations.</p>http://cvgmt.sns.it/paper/4117/The planning problem in Mean Field Games as regularized mass transporthttp://cvgmt.sns.it/paper/4116/P. J. Graber, A. R. Mészáros, F. Silva, D. Tonon.<p>In this paper, using variational approaches, we investigate the first order planning problem arising in the theory of mean field games. We show the existence and uniqueness of weak solutions of the problem in the case of a large class of Hamiltonians with arbitrary superlinear order of growth at infinity and local coupling functions. We require the initial and final measures to be merely summable. As an alternative way, we show that solutions of the planning problem can be approximated, via a $\Gamma$-convergence procedure, by solutions of standard mean field games with suitable penalized final couplings. In the same time (relying on the techniques developed recently by Graber and M\'esz\'aros), under stronger monotonicity and convexity conditions on the data, we obtain Sobolev estimates on the solutions of mean field games with general final couplings and the planning problem as well, both for space and time derivatives.</p>http://cvgmt.sns.it/paper/4116/Existence of a Lens–Shaped Cluster of Surfaces Self–Shrinking by Mean Curvaturehttp://cvgmt.sns.it/paper/4115/P. Baldi, E. Haus, C. Mantegazza.http://cvgmt.sns.it/paper/4115/The Dirichlet problem for the $p$-fractional Laplace equationhttp://cvgmt.sns.it/paper/4114/G. Palatucci.<p>We deal with a class of equations driven by nonlocal, possibly degenerate, integro-differential operators of differentiability order $s\in (0,1)$ and summability growth $p\in(1,\infty)$, whose model is the fractional $p$-Laplacian operator with measurable coefficients. We review several recent results for the corresponding weak solutions<i>supersolutions, as comparison principles, a priori bounds, lower semicontinuity, boundedness, H\"older continuity up to the boundary, and many others.We then discuss the good definition of $(s,p)$-superharmonic functions, and the nonlocal counterpart of the Perron method in nonlinear Potential Theory, together with various related results. We briefly mention some basic results for the obstacle problem for nonlinear integro-differential equations. Finally, we present the connection amongst the fractional viscosity solutions, the weak solutions and the aforementioned $(s,p)$-superharmonic functions, together with other important results for this class of equations when involving general measure data, and a surprising fractional version of the Gehring lemma.</p><p>We sketch the corresponding proofs of some of the results presented here, by especially underlining the development of new fractional localization techniques and other recent tools. Various open problems are listed throughout the paper.</i></p>http://cvgmt.sns.it/paper/4114/Intrinsic Differentiability and Intrinsic Regular Surfaces in Carnot Groupshttp://cvgmt.sns.it/paper/4113/D. Di Donato.<p> A Carnot group G is a connected, simply connected, nilpotent Lie group withstratified Lie algebra. Intrinsic regular surfaces in Carnot groups play thesame role as C<sup>1</sup> surfaces in Euclidean spaces. As in Euclidean spaces,intrinsic regular surfaces can be locally defined in different ways: e.g. asnon critical level sets or as continuously intrinsic differentiable graphs. Theequivalence of these natural definitions is the problem that we are studying.Precisely our aim is to generalize some results proved by Ambrosio, SerraCassano, Vittone valid in Heisenberg groups to the more general setting ofCarnot groups.</p>http://cvgmt.sns.it/paper/4113/A Liouville Theorem for Superlinear Heat Equations on Riemannian Manifoldshttp://cvgmt.sns.it/paper/4112/D. Castorina, C. Mantegazza, B. Sciunzi.<p>We study the triviality of the solutions of weighted superlinear heat equations on Riemannian manifolds with nonnegative Ricci tensor. We prove a Liouville-type theorem for solutions bounded from below with nonnegative initial data, under an integral growth condition on the weight.</p>http://cvgmt.sns.it/paper/4112/BV Functions in Metric Measure Spaces: Traces and Integration by Parts Formulæhttp://cvgmt.sns.it/paper/4110/V. Buffa.http://cvgmt.sns.it/paper/4110/A characterization of convex calibrable sets in $\mathbb R^N$ with respect to anisotropic normshttp://cvgmt.sns.it/paper/4109/V. Caselles, A. Chambolle, S. Moll, M. Novaga.<p>A set is called "calibrable" if its characteristic function is an eigenvector of the subgradient of the total variation. The main purpose of this paper is to characterize the $\phi$-calibrability of bounded convex sets in $\mathbb R^N$, with respect to a norm $\phi$ (called anisotropy), by the anisotropic mean $\phi$-curvature of its boundary. It extends to the anisotropic and crystalline cases the known analogous results in the Euclidean case. As a by-product of our analysis we prove that any convex body $C$ satisfying a $\phi$-ball condition contains a convex $\phi$-calibrable set $K$ such that, for any $V\in [<br>K<br>,<br>C<br>]$, the subset of $C$ of volume $V$ which minimizes the $\phi$-perimeter is unique and convex. We also describe the anisotropic total variation flow with initial data by the characteristic function of a bounded convex set.</p>http://cvgmt.sns.it/paper/4109/Front propagation in infinite cylinders II. The sharp reaction zone limithttp://cvgmt.sns.it/paper/4108/C. Muratov, M. Novaga.<p>This paper applies the variational approach developed in part I ofthis work to a singular limit of reaction-diffusion-advection equationswhich arise in combustion modeling. We first establish existence, uniqueness,monotonicity, asymptotic decay, and the associated free boundaryproblem for special traveling wave solutions which are minimizers of theconsidered variational problem in the singular limit. We then show thatthe speed of the minimizers of the approximating problems converges tothe speed of the minimizer of the singular limit. Also, after an appropriatetranslation the minimizers of the approximating problems convergestrongly on compacts to the minimizer of the singular limit. In addition,we obtain matching upper and lower bounds for the speed of the minimizersin the singular limit in terms of a certain area-type functional forsmall curvatures of the free boundary. The conclusions of the analysis areillustrated by a number of numerical examples.</p>http://cvgmt.sns.it/paper/4108/Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo's lemmahttp://cvgmt.sns.it/paper/4100/D. Spector, J. Van Schaftingen.<p>We prove a family of Sobolev inequalities of the form \[ \Vert u \Vert_{L^{\frac{n}{n-1}, 1} (\mathbb{R}^n,V)} \le C \Vert A (D) u \Vert_{L^1 (\mathbb{R}^n,E)} \]where $A (D) : C^\infty_c (\mathbb{R}^n, V) \to C^\infty_c (\mathbb{R}^n, E)$ is a vector first-order homogeneous linear differential operator with constant coefficients, $u$ is a vector field on $\mathbb{R}^n$ and $L^{\frac{n}{n - 1}, 1} (\mathbb{R}^{n})$ is a Lorentz space. These new inequalities imply in particular the extension of the classical Gagliardo--Nirenberg inequality to Lorentz spaces originally due to Alvino and a sharpening of an inequality in terms of the deformation operator by Strauss (Korn--Sobolev inequality) on the Lorentz scale. The proof relies on a nonorthogonal application of the Loomis--Whitney inequality and Gagliardo's lemma.</p>http://cvgmt.sns.it/paper/4100/On uniform measures in the Heisenberg grouphttp://cvgmt.sns.it/paper/4099/V. Chousionis, V. Magnani, J. T. Tyson.<p>We initiate a classification of uniform measures in the first Heisenberg group ℍ equipped with the Korányi metric dH, that represents the first example of a noncommutative stratified group equipped with a homogeneous distance. We prove that 1-uniform measures are proportional to the spherical 1-Hausdorff measure restricted to an affine horizontal line, while 2-uniform measures are proportional to spherical 2-Hausdorff measure restricted to an affine vertical line. It remains an open question whether 3-uniform measures are proportional to the restriction of spherical 3-Hausdorff measure to an affine vertical plane. We establish this conclusion in case the support of the measure is a vertically ruled surface. Along the way, we derive asymptotic formulas for the measures of small extrinsic balls in (ℍ,dH) intersected with smooth submanifolds. The coefficients in our power series expansions involve intrinsic notions of curvature associated to smooth curves and surfaces in ℍ.</p>http://cvgmt.sns.it/paper/4099/The Onsager Theoremhttp://cvgmt.sns.it/paper/4098/C. De Lellis.<p>In his famous 1949 paper on hydrodinamic turbulence, Lars Osanger advanced a remarkable conjecture on the energy conservation of weak solutions to the Euler equations: all H\"older continuous solutions with H\"older exponent strictly largerthan $\frac{1}{3}$ preserves the kinetic energy, while there are H\"older continuous solutions with any exponent strictly smaller than $\frac{1}{3}$ which do not preserve the kinetic energy. </p><p>While the first statement was proved by Constantin, E and Titi in 1994, the second was proved only recently by P. Isett buildingupon previous works of L\'aszl\'o Sz\'ekelyhidi Jr. and the author. This paper is a survey on the proof of the conjecture and on several other related discoveries which have been made in the last few years.</p>http://cvgmt.sns.it/paper/4098/On a question by Corson about point-finite coveringshttp://cvgmt.sns.it/paper/4097/A. Marchese, C. Zanco.<p>We answer in the affirmative the following question raised by H. H. Corson in 1961: "Is it possible to cover every Banach space X by bounded convex sets with nonempty interior in such a way that no point of X belongs to infinitely many of them?" Actually we show the way to produce in every Banach space X a bounded convex tiling of order 2, i.e. a covering of X by bounded convex closed sets with nonempty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles.</p>http://cvgmt.sns.it/paper/4097/Emergence of non-trivial minimizers for the three-dimensional Ohta-Kawasaki energyhttp://cvgmt.sns.it/paper/4096/H. Knüpfer, C. Muratov, M. Novaga.<p> This paper is concerned with the diffuse interface Ohta-Kawasaki energy inthree space dimensions, in a periodic setting, in the parameter regimecorresponding to the onset of non-trivial minimizers. We identify the scalingin which a sharp transition from asymptotically trivial to non-trivialminimizers takes place as the small parameter characterizing the width of theinterfaces between the two phases goes to zero, while the volume fraction ofthe minority phases vanishes at an appropriate rate. The value of the thresholdis shown to be related to the optimal binding energy solution of Gamow's liquiddrop model of the atomic nucleus. Beyond the threshold the average volumefraction of the minority phase is demonstrated to grow linearly with thedistance to the threshold. In addition to these results, we establish a numberof properties of the minimizers of the sharp interface screened Ohta-Kawasakienergy in the considered parameter regime. We also establish rather tight upperand lower bounds on the value of the transition threshold.</p>http://cvgmt.sns.it/paper/4096/A trace theorem for Martinet-type vector fieldshttp://cvgmt.sns.it/paper/4095/D. Gerosa, R. Monti, D. Morbidelli.<p>We prove a trace theorem of Besov typefor a subelliptic gradient of Martinet type.</p>http://cvgmt.sns.it/paper/4095/John and uniform domains in generalized Siegel boundarieshttp://cvgmt.sns.it/paper/4094/R. Monti, D. Morbidelli.<p>Given the pair of vector fields</p><p>$X=\partial_x+z ^{2m}y\partial_t$ </p><p>and </p><p>$Y=\partial_y-z^{2m}x \partial_t$</p><p>where $(x,y,t)= (z,t)\in\R^3=\C\times\R$, we give a condition on a bounded domain $\Omega\subset\R^3$ which ensures that $\Omega$ is an $(\epsilon,\delta)$-domain for the Carnot-Carath\'eodory metric. We alsoanalyze the Ahlfors regularity of the natural surface measure induced on $\partial \Omega$by the vector fields.</p>http://cvgmt.sns.it/paper/4094/Stable hypersurfaces in the complex projective spacehttp://cvgmt.sns.it/paper/4093/E. Battaglia, R. Monti, A. Righini.<p>We characterize the sphere with radius tan<sup>2</sup> r = 2n+1 in the complex projective space CP<sup>n</sup> as the unique stable hypersurface subject to certain bounds on the curvatures.</p>http://cvgmt.sns.it/paper/4093/Dimensional estimates and rectifiability for measures satisfying linear PDE constraintshttp://cvgmt.sns.it/paper/4092/A. Arroyo-Rabasa, G. De Philippis, J. Hirsch, F. Rindler.<p>We establish the rectifiability of measures satisfying a linear PDEconstraint. The obtained rectifiability dimensions are optimal for many usualPDE operators (including all first-order and all second-order operators). Ourgeneral theorem provides a new proof of the rectifiability results forfunctions of bounded variations and functions of bounded deformation. Fordivergence-free tensors we obtain refinements and new proofs of several knownresults on the rectifiability of varifolds and defect measures.</p>http://cvgmt.sns.it/paper/4092/Asymptotic spherical shapes in some spectral optimization problemshttp://cvgmt.sns.it/paper/4091/D. Mazzoleni, B. Pellacci, G. Verzini.<p>We study the optimization of the positive principal eigenvalue of an indefinite weighted problem,associated with the Neumann Laplacian in a box $\Omega\subset\mathbb{R}^N$, which arises inthe investigation of the survival threshold in population dynamics.When trying to minimize such eigenvalue with respect to the weight, one is lead toconsider a shape optimization problem, which is known to admit no spherical optimalshapes (despite some previously stated conjectures).We investigate whether spherical shapes can be recovered in some singular perturbationlimit. More precisely we show that, whenever the negative part of the weight diverges, the above shape optimization problem approaches in the limitthe so called spectral drop problem, which involves the minimization of the firsteigenvalue of the mixed Dirichlet-Neumann Laplacian. We prove that, for suitable choicesof the box $\Omega$, the optimal shapes for this second problem are indeed spherical;moreover, for general $\Omega$, we show that small volume spectral drops areasymptotically spherical, with center at points of $\partial\Omega$ having large mean curvature.</p>http://cvgmt.sns.it/paper/4091/Starshaped and convex sets in Carnot groups and in the geometries of vector fields.http://cvgmt.sns.it/paper/4090/F. Dragoni, D. Filali.http://cvgmt.sns.it/paper/4090/