CVGMT Papershttp://cvgmt.sns.it/papers/en-usFri, 20 Jul 2018 21:51:04 +0000Existence and regularity results for minimal sets; Plateau problemhttp://cvgmt.sns.it/paper/3967/cavallotto.<p>Solving the Plateau problem means to find the surface with minimal area amongall surfaces with a given boundary. Part of the problem actually consists ofgiving a suitable definition to the notions of 'surface', 'area' and'boundary'. In our setting the considered objects are sets whose Hausdorff areais locally finite. The sliding boundary condition is given in term of a oneparameter family of compact deformations which allows the boundary of thesurface to moove along a closed set. The area functional is related tocapillarity and free-boundary problems, and is a slight modification of theHausdorff area. We focused on minimal boundary cones; that is to say tangentcones on boundary points of sliding minimal surfaces. In particular we studiedcones contained in an half-space and whose boundary can slide along thebounding hyperplane. After giving a classification of one-dimensional minimalcones in the half-plane we provided four new two-dimensional minimal cones inthe three-dimensional half space (which cannot be obtained as the Cartesianproduct of the real line with one of the previous cones). We employed thetechnique of paired calibrations and in one case could also generalise it tohigher dimension. In order to prove that the provided list of minimal cones iscomplete, we started the classification of cones satisfying the necessaryconditions for the minimality, and with numeric simulations we obtained bettercompetitors for these new candidates.</p>http://cvgmt.sns.it/paper/3967/A $C^m$ Whitney extension theorem for horizontal curves in the Heisenberg grouphttp://cvgmt.sns.it/paper/3966/A. Pinamonti, G. Speight, S. Zimmerman.<p>We characterize those mappings from a compact subset of $\mathbb{R}$ intothe Heisenberg group $\mathbb{H}^n$ which can be extended to a $C^m$ horizontal curve in $\mathbb{H}^n$. The characterization combines the classical Whitney conditions with anestimate comparing changes in the vertical coordinate with those predicted bythe Taylor series of the horizontal coordinates.</p>http://cvgmt.sns.it/paper/3966/Nonexistence for hyperbolic problems on riemannian manifoldshttp://cvgmt.sns.it/paper/3965/D. D. Monticelli, F. Punzo, M. Squassina.<p>We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.</p>http://cvgmt.sns.it/paper/3965/Nonlinear Dirac Equation On Graphs With Localized Nonlinearities: Bound States And Nonrelativistic Limithttp://cvgmt.sns.it/paper/3964/W. Borrelli, R. Carlone, L. Tentarelli.<p>In this paper we study the nonlinear Dirac (NLD) equation on noncompactmetric graphs with localized Kerr nonlinearities, in the case of Kirchhoff-typeconditions at the vertices. Precisely, we discuss existence and multiplicity ofthe bound states (arising as critical points of the NLD action functional) andwe prove that, in the L 2-subcritical case, they converge to the bound statesof the NLS equation in the nonrelativistic limit.</p>http://cvgmt.sns.it/paper/3964/Blow-up analysis of a nonlocal Liouville-type equationhttp://cvgmt.sns.it/paper/3963/F. Da Lio, L. Martinazzi, T. RiviÃ¨re.<p> In this paper we perform a blow-up and quantization analysis of the followingnonlocal Liouville-type equation \begin{equation}(-\Delta)<sup>\frac12</sup> u= \kappae<sup>u</sup>-1~\mbox{in $S^1$,} \end{equation} where $(-\Delta)^\frac{1}{2}$ stands forthe fractional Laplacian and $\kappa$ is a bounded function. We interpret theabove equation as the prescribed curvature equation to a curve in conformalparametrization. We also establish a relation between this equation and theanalogous equation in $\mathbb{R}$ \begin{equation} (-\Delta)<sup>\frac{1}{2}</sup> u =Ke<sup>u</sup> \quad \text{in }\mathbb{R}, \end{equation} with$K$ bounded on $\mathbb{R}$.</p>http://cvgmt.sns.it/paper/3963/The non-parametric problem of Plateau in arbitrary codimensionhttp://cvgmt.sns.it/paper/3962/Luca M. Martinazzi.<p> We study the problem of finding a minimal graph with prescribed boundary datain arbitrary dimension and codimension. Existence, uniqueness, stability andregularity are treated. We first present the well-known results for codimensionone: Jenkins-Serrin's existence theorem, convexity properties of the area whichgive uniqueness and stability and De Giorgi's theorem for regularity. In highercodimension we first discuss the counterexamples of Lawson and Osserman. Thenwe present the recent results of Mu-Tao Wang: we use the mean curvature flow toshow existence for small boundary data and we prove a new Bernstein theorem(also due to Mu-Tao Wang) which holds for every area-decreasing minimal graph.We finally deduce from the Bernstein theorem that an area-decreasing minimalgraph is always smooth.</p>http://cvgmt.sns.it/paper/3962/Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension threehttp://cvgmt.sns.it/paper/3961/T. Jin, A. Maalaoui, L. Martinazzi, J. Xiong.<p> We study conformal metrics on $R^3$, i.e., metrics of the form$g_u=e^{2u}<br>dx<br>^2$, which have constant $Q$-curvature and finite volume. Thisis equivalent to studying the non-local equation $$ (-\Delta)<sup>\frac32</sup> u = 2e<sup>{3u}$$</sup> in $R^3$ $$V:=\int<sub>{\mathbb{R}</sub><sup>3}e</sup><sup>{3u}dx<\infty,$$</sup> where $V$ is thevolume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to thenon-local framework, we show the existence of a large class of such metrics,particularly for $V\le 2\pi^2=<br>S^3<br>$. Inspired by previous works of C-S. Linand L. Martinazzi, who treated the analogue cases in even dimensions, weclassify such metrics based on their behavior at infinity.</p>http://cvgmt.sns.it/paper/3961/Critical points of the Moser-Trudinger functionalhttp://cvgmt.sns.it/paper/3960/F. De Marchis, A. Malchiodi, L. Martinazzi.<p> On a smooth bounded 2-dimensional domain $\Omega$ we study the heat flow$u_t=\Delta u +\lambda (t)ue^{u^2}$ ($\lambda(t)$ is such that $d/dt|u(t,\cdot)|_{H^1_0}=0$) introduced by T. Lamm, F. Robert and M. Struwe toinvestigate the Moser-Trudinger functional $E(v)=\int_{\Omega} (e^{v^2}-1)dx,v\in H^1_0(\Omega).$ We prove that if $u$ blows-up as $t\to\infty$ and if$E(u(t,\cdot))$ remains bounded, then for a sequence $t_k\to\infty$ we have$u(t_k,\cdot)\rightharpoonup 0$ in $H^1_0$ and $\<br>u(t_k,\cdot)\<br>_{H^1_0}^2\to4\pi L$ for an integer $L\ge 1$. We couple these results with a topological technique to prove that if$\Omega$ is not contractible, then for every $0<\Lambda\in \mathbb{R} \setminus4 \pi \mathbb{N}$ the functional $E$ constrained to $M_\Lambda=\{v\inH^1_0(\Omega):|v|_{H^1_0}^2=\Lambda \}$ has a positive critical point. Weprove that when $\Omega$ is the unit ball and $\Lambda$ is large enough, then$E<br>_{M_\Lambda}$ has no positive critical points, hence showing that thetopological assumption on $\Omega$ is natural.</p>http://cvgmt.sns.it/paper/3960/Conformal metrics on R^{2m} with constant Q-curvature and large volumehttp://cvgmt.sns.it/paper/3959/L. Martinazzi.<p> We study conformal metrics on R<sup>{2m}</sup> with constant Q-curvature and finitevolume. When m=3 we show that there exists V<b> such that for any V\in<a href='V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature ofS^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case,treated by C-S. Lin. We also prove that when $m$ is odd and greater than 1,there is a constant V_m>\vol (S^{2m}) such that for every V\in (0,V_m'>V*,\infty) there is a conformal metric g on R^{6} with Q_g = Q-curvature ofS^6, and vol(g)=V. This is in sharp contrast with the four-dimensional case,treated by C-S. Lin. We also prove that when $m$ is odd and greater than 1,there is a constant V_m>\vol (S^{2m}) such that for every V\in (0,V_m</a> there isa conformal metric g on R<sup>{2m}</sup> with Q<sub>g</sub> = Q-curvature of S<sup>{2m},</sup> vol(g)=V. Thisextends a result of A. Chang and W-X. Chen. When m is even we prove a similarresult for conformal metrics of negative Q-curvature.</b></p>http://cvgmt.sns.it/paper/3959/Blow-up behaviour of a fractional Adams-Moser-Trudinger type inequality in odd dimensionhttp://cvgmt.sns.it/paper/3958/A. Maalaoui, L. Martinazzi, A. Schikorra.<p> Given a smoothly bounded domain $\Omega\Subset\mathbb{R}^n$ with $n\ge 1$odd, we study the blow-up of bounded sequences $(u_k)\subsetH^\frac{n}{2}_{00}(\Omega)$ of solutions to the non-local equation$$(-\Delta)<sup>\frac</sup> n2 u<sub>k=\lambda</sub><sub>k</sub> u<sub>ke</sub><sup>{\frac</sup> n2 u<sub>k</sub><sup>2}\quad</sup> \text{in}\Omega,$$ where $\lambda_k\to\lambda_\infty \in [0,\infty)$, and $H^{\fracn2}_{00}(\Omega)$ denotes the Lions-Magenes spaces of functions $u\inL^2(\mathbb{R}^n)$ which are supported in $\Omega$ and with$(-\Delta)^\frac{n}{4}u\in L^2(\mathbb{R}^n)$. Extending previous works ofDruet, Robert-Struwe and the second author, we show that if the sequence$(u_k)$ is not bounded in $L^\infty(\Omega)$, a suitably rescaled subsequence$\eta_k$ converges to the function$\eta_0(x)=\log\left(\frac{2}{1+<br>x<br>^2}\right)$, which solves the prescribednon-local $Q$-curvature equation $$(-\Delta)<sup>\frac</sup> n2 \eta=(n-1)!e<sup>{n\eta}\quad</sup> \text{in }\mathbb{R}<sup>n$$</sup> recently studied by DaLio-Martinazzi-Rivi\`ere when $n=1$, Jin-Maalaoui-Martinazzi-Xiong when $n=3$,and Hyder when $n\ge 5$ is odd. We infer that blow-up can occur only if$\Lambda:=\limsup_{k\to \infty}\<br>(-\Delta)^\frac n4 u_k\<br>_{L^2}^2\ge\Lambda_1:= (n-1)!<br>S^n<br>$.</p>http://cvgmt.sns.it/paper/3958/A fractional Moser-Trudinger type inequalitiy in one dimension and its critical pointshttp://cvgmt.sns.it/paper/3957/S. Iula, A. Maalaoui, L. Martinazzi.<p> We study a sharp fractional Moser-Trudinger type inequality in dimension 1,its compactness properties and the critical points of a functional associetedto the inequality.</p>http://cvgmt.sns.it/paper/3957/The nonlocal Liouville-type equation in $\mathbb{R}$ and conformal immersions of the disk with boundary singularitieshttp://cvgmt.sns.it/paper/3955/F. Da Lio, L. Martinazzi.<p> In this paper we perform a blow-up and quantization analysis of thefractional Liouville equation in dimension $1$. More precisely, given asequence $u_k :\mathbb{R} \to \mathbb{R}$ of solutions to \begin{equation} (-\Delta)<sup>\frac{1}{2}</sup> u<sub>k</sub> =K<sub>ke</sub><sup>{u</sup><sub>k}\quad</sub> \text{in }\mathbb{R},\end{equation} with $K_k$ bounded in $L^\infty$ and $e^{u_k}$ bounded in $L^1$uniformly with respect to $k$, we show that up to extracting a subsequence$u_k$ can blow-up at (at most) finitely many points $B=\{a_1,\dots, a_N\}$ andeither (i) $u_k\to u_\infty$ in $W^{1,p}_{loc}(\mathbb{R}\setminus B)$ and$K_ke^{u_k} \stackrel{*}{\rightharpoondown} K_\infty e^{u_\infty}+ \sum_{j=1}^N\pi \delta_{a_j}$, or (ii) $u_k\to-\infty$ uniformly locally in$\mathbb{R}\setminus B$ and $K_k e^{u_k}\stackrel{*}{\rightharpoondown}\sum_{j=1}^N \alpha_j \delta_{a_j}$ with $\alpha_j\ge \pi$ for every $j$. Thisresult, resting on the geometric interpretation and analysis provided in arecent collaboration of the authors with T. Rivi\`ere and on a classical workof Blank about immersions of the disk into the plane, is a fractionalcounterpart of the celebrated works of Br\'ezis-Merle and Li-Shafrir on the$2$-dimensional Liouville equation, but providing sharp quantization estimates($\alpha_j=\pi$ and $\alpha_j\ge \pi$) which are not known in dimension $2$under the weak assumption that $(K_k)$ be bounded in $L^\infty$ and is allowedto change sign.</p>http://cvgmt.sns.it/paper/3955/Fractional Adams-Moser-Trudinger type inequalitieshttp://cvgmt.sns.it/paper/3956/L. Martinazzi.<p> Extending several works, we prove a general Adams-Moser-Trudinger typeinequality for the embedding of Bessel-potential spaces $\tildeH^{\frac{n}{p},p}(\Omega)$ into Orlicz spaces for an arbitrary domain$\Omega\subset \mathbb{R}^n$ with finite measure. In particular we prove$$\sup<sub>{u\in</sub> \tilde H<sup>{\frac{n}{p},p}</sup>(\Omega),\;\<br>(-\Delta)<sup>{\frac{n}{2p}}u\<br></sup><sub>{L</sub><sup>{p}</sup>(\Omega)}\leq1}\int<sub>{\Omega}e</sub><sup>{\alpha</sup><sub>{n,p}</sub> <br>u<br><sup>\frac{p}{p</sup>-1}}dx \leq c<sub>{n,p}<br>\Omega<br>,</sub> $$for a positive constant $\alpha_{n,p}$ whose sharpness we also prove. Wefurther extend this result to the case of Lorentz-spaces (i.e.$(-\Delta)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple, as they useGreen functions for fractional Laplace operators and suitable cut-offprocedures to reduce the fractional results to the sharp estimate on the Rieszpotential proven by Adams and its generalization proven by Xiao and Zhai. Wealso discuss an application to the problem of prescribing the $Q$-curvature andsome open problems.</p>http://cvgmt.sns.it/paper/3956/The Moser-Trudinger inequality and its extremals on a disk via energy estimateshttp://cvgmt.sns.it/paper/3954/G. Mancini, L. Martinazzi.<p> We study the Dirichlet energy of non-negative radially symmetric criticalpoints $u_\mu$ of the Moser-Trudinger inequality on the unit disc in$\mathbb{R}^2$, and prove that it expands as$$4\pi+\frac{4\pi}{\mu<sup>{4}}+o</sup>(\mu<sup>{</sup>-4})\le \int<sub>{B</sub><sub>1}<br>\nabla</sub> u<sub>\mu<br></sub><sup>2dx\le</sup>4\pi+\frac{6\pi}{\mu<sup>{4}}+o</sup>(\mu<sup>{</sup>-4}),\quad \text{as }\mu\to\infty,$$ where$\mu=u_\mu(0)$ is the maximum of $u_\mu$. As a consequence, we obtain a newproof of the Moser-Trudinger inequality, of the Carleson-Chang result about theexistence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicityresult in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of theMoser-Trudinger functional. We explicitly identify the critical level ofperturbation for which, although the perturbed Moser-Trudinger inequality stillholds, the energy of its critical points converges to $4\pi$ from below. Weexpect, in some of these cases, that the existence of extremals does not hold,nor the existence of critical points in the supercritical regime.</p>http://cvgmt.sns.it/paper/3954/Large blow-up sets for the prescribed Q-curvature equation in the Euclidean spacehttp://cvgmt.sns.it/paper/3953/A. Hyder, S. Iula, L. Martinazzi.<p> Let $m\ge 2$ be an integer. For any open domain$\Omega\subset\mathbb{R}^{2m}$, non-positive function $\varphi\inC^\infty(\Omega)$ such that $\Delta^m \varphi\equiv 0$, and bounded sequence$(V_k)\subset L^\infty(\Omega)$ we prove the existence of a sequence offunctions $(u_k)\subset C^{2m-1}(\Omega)$ solving the Liouville equation oforder $2m$ $$(-\Delta)<sup>m</sup> u<sub>k</sub> = V<sub>ke</sub><sup>{2mu</sup><sub>k}\quad</sub> \text{in }\Omega, \quad\limsup<sub>{k\to\infty}</sub> \int<sub>\Omega</sub> e<sup>{2mu</sup><sub>k}dx<\infty,$$</sub> and blowing up exactlyon the set $S_{\varphi}:=\{x\in \Omega:\varphi(x)=0\}$, i.e.$$\lim<sub>{k\to\infty}</sub> u<sub>k</sub>(x)=+\infty \text{ for }x\in S<sub>{\varphi}</sub> \text{ and}\lim<sub>{k\to\infty}</sub> u<sub>k</sub>(x)=-\infty \text{ for }x\in \Omega\setminusS<sub>{\varphi},$$</sub> thus showing that a result of Adimurthi, Robert and Struwe issharp. We extend this result to the boundary of $\Omega$ and to the case$\Omega=\mathbb{R}^{2m}$. Several related problems remain open.</p>http://cvgmt.sns.it/paper/3953/Gluing metrics with prescribed $Q$-curvature and different asymptotic behaviour in dimension $6$http://cvgmt.sns.it/paper/3952/A. Hyder, L. Martinazzi.<p> We show a new example of blow-up behaviour for the prescribed $Q$-curvatureequation in dimension $6$, namely given a sequence $(V_k)\subsetC^0(\mathbb{R}^6)$ suitably converging we construct a sequence $(u_k)$ ofradially symmetric solutions to the equation $$(-\Delta)<sup>3</sup> u<sub>k=V</sub><sub>k</sub> e<sup>{6</sup> u<sub>k}</sub>\quad \text{in }\mathbb{R}<sup>6,$$</sup> with $u_k$ blowing up at the origin and on asphere. We also prove sharp blow-up estimates.</p>http://cvgmt.sns.it/paper/3952/Optimal Lipschitz criteria and local estimates for non-uniformly elliptic problemshttp://cvgmt.sns.it/paper/3951/L. Beck, G. Mingione.<p> We report on new techniques and results in the regularity theory of generalnon-uniformly elliptic variational integrals. By means of a new potentialtheoretic approach we reproduce, in the non-uniformly elliptic setting, theoptimal criteria for Lipschitz continuity known in the uniformly elliptic oneand provide a unified approach between non-uniformly and uniformly ellipticproblems.</p>http://cvgmt.sns.it/paper/3951/Spectral partitions for Sturm-Liouville problemshttp://cvgmt.sns.it/paper/3950/P. Tilli, D. Zucco.<p>We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via \Gamma-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm-Liouville operator.</p>http://cvgmt.sns.it/paper/3950/Almgren's center manifold in a simple settinghttp://cvgmt.sns.it/paper/3949/C. De Lellis.<p>We aim at explaining the most basic ideas underlying two fundamental results in the regularity theory of area minimizing oriented surfaces: De Giorgi's celebrated $\varepsilon$-regularity theorem and Almgren's center manifold. Both theorems will be proved in a very simplified situation, which however allows to illustrate some of the most important PDE estimates.</p><p>Lectures held at Park City 9-13 July 2018</p>http://cvgmt.sns.it/paper/3949/Rigidity of positively curved shrinking Ricci solitons in dimension fourhttp://cvgmt.sns.it/paper/3948/G. Catino.<p>We classify four-dimensional shrinking Ricci solitons satisfying $Sec \geq \frac{1}{48} R$, where $Sec$ and $R$ denote the sectional and the scalar curvature, respectively. They are isometric to either $\mathbb{R}^{4}$ (and quotients), $\mathbb{S}^{4}$, $\mathbb{RP}^{4}$ or $\mathbb{CP}^{2}$ with their standard metrics.</p>http://cvgmt.sns.it/paper/3948/