Authors:Valentina Franceschi Abstract: We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional “one-dimensional” constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters. PubDate: 2017-07-08 DOI: 10.1007/s00526-017-1198-5 Issue No:Vol. 56, No. 4 (2017)

Authors:Farhan Abedin; Cristian E. Gutiérrez Abstract: The purpose of this paper is to present an iterative scheme to find approximate solutions, to any preset degree of accuracy, for a class of generated Jacobian equations introduced in Trudinger (Discrete Contin Dyn Syst 34(4): 1663–1681 2014). A proof of an upper bound on the number of iteration steps, without assuming any condition of Ma-Trudinger-Wang type, is presented. Applications to optimal mass transport and to the parallel reflector problem in optics are also provided. PubDate: 2017-07-07 DOI: 10.1007/s00526-017-1200-2 Issue No:Vol. 56, No. 4 (2017)

Authors:Manh Hong Duong; Agnes Lamacz; Mark A. Peletier; Upanshu Sharma Abstract: In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with non-dissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov–Fokker–Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. PubDate: 2017-06-28 DOI: 10.1007/s00526-017-1186-9 Issue No:Vol. 56, No. 4 (2017)

Authors:YanYan Li; Luc Nguyen Abstract: We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem. PubDate: 2017-06-19 DOI: 10.1007/s00526-017-1192-y Issue No:Vol. 56, No. 4 (2017)

Authors:Zhiqin Lu; Yi Wang Abstract: In this paper, we study the ends of a locally conformally flat complete manifold with finite total Q-curvature. We prove that for such a manifold, the integral of the Q-curvature equals an integral multiple of a dimensional constant \(c_n\) , where \(c_n\) is the integral of the Q-curvature on the unit n-sphere. It provides further evidence that the Q-curvature on a locally conformally flat manifold controls geometry as the Gaussian curvature does in two dimension. PubDate: 2017-06-16 DOI: 10.1007/s00526-017-1189-6 Issue No:Vol. 56, No. 4 (2017)

Authors:Ze Li; Lifeng Zhao Abstract: In this paper, we study the asymptotic behaviors of finite energy solutions to the Landau–Lifshitz flows from \(\mathbb {R}^2\) into Kähler manifolds. First, we prove that the solution with initial data below the critical energy converges to a constant map in the energy space as \(t\rightarrow \infty \) for the compact Riemannian surface targets. In particular, when the target is a two dimensional sphere, we prove that the solution to the Landau–Lifshitz–Gilbert equation with initial data having an energy below \(4\pi \) converges to some constant map in the energy space. The proof bases on the method of induction on energy and geometric renormalizations. Second, for general compact Kähler manifolds and initial data of an arbitrary finite energy, we obtain a bubbling theorem analogous to the Struwe’s results on the heat flows. PubDate: 2017-06-10 DOI: 10.1007/s00526-017-1182-0 Issue No:Vol. 56, No. 4 (2017)

Authors:Michinori Ishiwata; Rolando Magnanini; Hidemitsu Wadade Abstract: Let \(1\le p\le \infty \) . We show that a function \(u\in C(\mathbb R^N)\) is a viscosity solution to the normalized p-Laplace equation \(\Delta _p^n u(x)=0\) if and only if the asymptotic formula $$\begin{aligned} u(x)=\mu _p(\varepsilon ,u)(x)+o(\varepsilon ^2) \end{aligned}$$ holds as \(\varepsilon \rightarrow 0\) in the viscosity sense. Here, \(\mu _p(\varepsilon ,u)(x)\) is the p-mean value of u on \(B_\varepsilon (x)\) characterized as a unique minimizer of $$\begin{aligned} \Vert u-\lambda \Vert _{L^p(B_\varepsilon (x))} \end{aligned}$$ with respect to \(\lambda \in {\mathbb {R}}\) . This kind of asymptotic mean value property (AMVP) extends to the case \(p=1\) previous (AMVP)’s obtained when \(\mu _p(\varepsilon ,u)(x)\) is replaced by other kinds of mean values. The natural definition of \(\mu _p(\varepsilon ,u)(x)\) makes sure that this is a monotonic and continuous (in the appropriate topology) functional of u. These two properties help to establish a fairly general proof of (AMVP), that can also be extended to the (normalized) parabolic p-Laplace equation. PubDate: 2017-06-10 DOI: 10.1007/s00526-017-1188-7 Issue No:Vol. 56, No. 4 (2017)

Authors:Andrea Davini; Elena Kosygina Abstract: It was pointed out by P.-L. Lions, G. Papanicolaou, and S.R.S. Varadhan in their seminal paper (1987) that, for first order Hamilton–Jacobi (HJ) equations, homogenization starting with affine initial data implies homogenization for general uniformly continuous initial data. The argument makes use of some properties of the HJ semi-group, in particular, the finite speed of propagation. This property is lost for viscous HJ equations. In this paper we prove the above mentioned implication in both viscous and non-viscous cases. Our proof relies on a variant of Evans’s perturbed test function method. As an application, we show homogenization in the stationary ergodic setting for viscous and non-viscous HJ equations in one space dimension with non-convex Hamiltonians of specific form. The results are new in the viscous case. PubDate: 2017-06-08 DOI: 10.1007/s00526-017-1177-x Issue No:Vol. 56, No. 4 (2017)

Authors:Gabriele Mancini; Luca Martinazzi Abstract: We study the Dirichlet energy of non-negative radially symmetric critical points \(u_\mu \) of the Moser–Trudinger inequality on the unit disc in \(\mathbb {R}^2\) , and prove that it expands as $$\begin{aligned} 4\pi +\frac{4\pi }{\mu ^{4}}+o(\mu ^{-4})\le \int _{B_1} \nabla u_\mu ^2dx\le 4\pi +\frac{6\pi }{\mu ^{4}}+o(\mu ^{-4}),\quad \text {as }\mu \rightarrow \infty , \end{aligned}$$ where \(\mu =u_\mu (0)\) is the maximum of \(u_\mu \) . As a consequence, we obtain a new proof of the Moser–Trudinger inequality, of the Carleson–Chang result about the existence of extremals, and of the Struwe and Lamm–Robert–Struwe multiplicity result in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of the Moser–Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser–Trudinger inequality still holds, the energy of its critical points converges to \(4\pi \) from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime. PubDate: 2017-06-08 DOI: 10.1007/s00526-017-1184-y Issue No:Vol. 56, No. 4 (2017)

Authors:Marcel G. Clerc; Juan Diego Dávila; Michał Kowalczyk; Panayotis Smyrnelis; Estefania Vidal-Henriquez Abstract: We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parameters various defects are predicted by the model. In particular we show existence of a new type of topological defect which we call the shadow kink. Its local profile is described by the generalized Hastings and McLeod solutions of the second Painlevé equation (Claeys et al. in Ann Math 168(2):601–641, 2008; Hastings and McLeod in Arch Ration Mech Anal 73(1):31–51, 1980). As part of our analysis we give a new proof of existence of these solutions. PubDate: 2017-06-07 DOI: 10.1007/s00526-017-1187-8 Issue No:Vol. 56, No. 4 (2017)

Authors:Giovanni Anello; Francesca Faraci Abstract: We study the Dirichlet boundary value problem with 0-boundary data for the semilinear elliptic equation \(-\Delta u= (\lambda u^{s-1}-u^{r-1})\chi _{\{u>0\}}\) in a bounded domain \(\Omega \) , where \(0<r<s<1\) and \(\lambda \in (0,\infty )\) . In particular, for \(\lambda \) large enough, we prove the existence of at least two nonnegative solutions, one of which is positive, satisfies the Hopf’s boundary condition and corresponds to a local minimum of the energy functional. This paper is motivated by a recent result of the authors where the same conclusion was obtained for the case \(0<r\le 1< s<2\) . PubDate: 2017-06-06 DOI: 10.1007/s00526-017-1179-8 Issue No:Vol. 56, No. 4 (2017)

Authors:Gisella Croce; Giovanni Pisante Abstract: We consider the vectorial system $$\begin{aligned} {\left\{ \begin{array}{ll} Du \in \mathcal {O}(2), &{} \text{ a.e. } \text{ in }\,\;\Omega , \\ u=0, &{} \text{ on } \,\;\partial \Omega , \end{array}\right. } \end{aligned}$$ where \(\Omega \) is a subset of \(\mathbb R^2\) , \(u:\Omega \rightarrow \mathbb R^2\) and \(\mathcal {O}(2)\) is the orthogonal group of \(\mathbb R^2\) . We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appropriate weighted measure of some set of singularities of the gradient. PubDate: 2017-06-06 DOI: 10.1007/s00526-017-1185-x Issue No:Vol. 56, No. 4 (2017)

Authors:Patrick W. Dondl; Stephan Wojtowytsch Abstract: We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure \(\mu \) . For this purpose, we introduce a suitable notion of essentially uniform convergence. This mode of convergence is a natural generalisation of uniform convergence that precisely describes the convergence of phase fields in three dimensions. More in detail, we show that, in three space dimensions, points close to which the phase fields stay bounded away from a pure phase lie either in the support of the limiting mass measure \(\mu \) or contribute a positive amount to the limiting Willmore energy. Thus there can only be finitely many such points. As an application, we investigate the Hausdorff limit of level sets of sequences of phase fields with bounded energy. We also obtain results on boundedness and \(L^p\) -convergence of phase fields and convergence from outside the interval between the wells of a double-well potential. For minimisers of suitable energy functionals, we deduce uniform convergence of the phase fields from essentially uniform convergence. PubDate: 2017-06-05 DOI: 10.1007/s00526-017-1178-9 Issue No:Vol. 56, No. 4 (2017)

Authors:Tristan C. Collins; Sebastien Picard; Xuan Wu Abstract: We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if \(\Omega \) is a compact domain in \({\mathbb {R}}^{n}\) or \({\mathbb {C}}^n\) , then there exists a solution to the Dirichlet problem with right-hand side h(x) satisfying \( h(x) > (n-2)\frac{\pi }{2}\) and boundary data \(\varphi \) if and only if there exists a subsolution. PubDate: 2017-06-05 DOI: 10.1007/s00526-017-1191-z Issue No:Vol. 56, No. 4 (2017)

Authors:Leszek Gasiński; Nikolaos S. Papageorgiou Abstract: We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti–Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions. PubDate: 2017-06-03 DOI: 10.1007/s00526-017-1180-2 Issue No:Vol. 56, No. 3 (2017)

Authors:Yong Luo Abstract: In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo (arXiv:1211.4227v6) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in \(\mathbb {S}^5\) , and then we use this relation to prove a classification result for Willmore Legendrian spheres in \(\mathbb {S}^5\) . We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface in \(\mathbb {S}^5\) belongs to [0, 2], then it must be 0 and L is totally geodesic or 2 and L is a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Let \(\Sigma \) be a closed surface and \((M,\alpha ,g_\alpha ,J)\) a 5-dimensional Sasakian manifold with a contact form \(\alpha \) , an associated metric \(g_\alpha \) and an almost complex structure J. Assume that \(f:\Sigma \mapsto M\) is a Legendrian immersion. Then f is called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed if \((M,\alpha ,g_\alpha ,J)\) is a Sasakian Einstein manifold, in particular \(\mathbb {S}^5\) . PubDate: 2017-06-02 DOI: 10.1007/s00526-017-1183-z Issue No:Vol. 56, No. 3 (2017)

Authors:Patrick Guidotti Abstract: In the past few decades maximal regularity theory has successfully been applied to moving boundary problems. The basic idea is to reduce the system with varying domains to one in a fixed domain. This is done by a transformation, the so-called Hanzawa transformation, and yields a typically nonlocal and nonlinear coupled system of (evolution) equations. Well-posedness results can then often be established as soon as it is proved that the relevant linearization is the generator of an analytic semigroup or admits maximal regularity. To implement this program, it is necessary to somehow parametrize to space of boundaries/domains (typically the space of compact hypersurfaces \(\Gamma \) in \({\mathbb {R}}^n\) , in the Euclidean setting). This has traditionally been achieved by means of the already mentioned Hanzawa transformation. The approach, while successful, requires the introduction of a smooth manifold \(\Gamma _\infty \) close to the manifold \(\Gamma _0\) in which one cares to linearize. This prevents one to use coordinates in which \(\Gamma _0\) lies at their “center”. As a result formulæ tend to contain terms that would otherwise not be present were one able to linearize in a neighborhood emanating from \(\Gamma _0\) instead of from \(\Gamma _\infty \) . In this paper it is made use of flows (curves of diffeomorphisms) to obtain a general form of the relevant linearization in combination with an alternative coordinatization of the manifold of hypersurfaces, which circumvents the need for the introduction of a “phantom” reference manifold \(\Gamma _\infty \) by, in its place, making use of a “phantom geometry” on \(\Gamma _0\) . The upshot is a clear insight into the structure of the linearization, simplified calculations, and simpler formulæ for the resulting linear operators, which are useful in applications. PubDate: 2017-06-02 DOI: 10.1007/s00526-017-1181-1 Issue No:Vol. 56, No. 3 (2017)

Authors:Xiaoli Han; Jiayu Li; Jun Sun Abstract: In this paper we consider the compactness of \(\beta \) -symplectic critical surfaces in a Kähler surface. Let M be a compact Kähler surface and \(\Sigma _i\subset M\) be a sequence of closed \(\beta _i\) -symplectic critical surfaces with \(\beta _i\rightarrow \beta _0\in (0,\infty )\) . Suppose the quantity \(\int _{\Sigma _i}\frac{1}{\cos ^q\alpha _i}d\mu _i\) (for some \(q>4\) ) and the genus of \(\Sigma _{i}\) are bounded, then there exists a finite set of points \({{\mathcal {S}}}\subset M\) and a subsequence \(\Sigma _{i'}\) which converges uniformly in the \(C^l\) topology (for any \(l<\infty \) ) on compact subsets of \(M\backslash {{\mathcal {S}}}\) to a \(\beta _0\) -symplectic critical surface \(\Sigma \subset M\) , each connected component of \(\Sigma \setminus {{\mathcal {S}}}\) can be extended smoothly across \({{\mathcal {S}}}\) . PubDate: 2017-06-01 DOI: 10.1007/s00526-017-1175-z Issue No:Vol. 56, No. 3 (2017)

Authors:Flávio F. Cruz Abstract: In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, without assuming the convexity of the prescribed solution and using the theory of fully nonlinear elliptic equations. If the given data admits a suitable radial graph as a subsolution, then we prove that there exists a radial graph with constant curvature and realizing the prescribed boundary. As an application, it is proved that if \(\Omega \subset \mathbb {S}^n\) is a mean convex domain whose closure is contained in an open hemisphere of \(\mathbb {S}^n\) then, for \(0<R<n(n-1),\) there exists a radial graph of constant scalar curvature R and boundary \(\partial \Omega \) . PubDate: 2017-05-31 DOI: 10.1007/s00526-017-1176-y Issue No:Vol. 56, No. 3 (2017)