Authors:Hitoshi Ishii; Panagiotis E. Souganidis; Hung V. Tran Abstract: Abstract We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski–Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations. PubDate: 2017-10-13 DOI: 10.1007/s00526-017-1240-7 Issue No:Vol. 56, No. 6 (2017)

Authors:Wenjing Chen; Shengbing Deng; Seunghyeok Kim Abstract: Abstract Let \((X, g^+)\) be an asymptotically hyperbolic manifold and \((M, [\hat{h}])\) be its conformal infinity. We construct positive clustered solutions to low-order perturbations of \(\gamma \) -Yamabe equations ( \(0< \gamma < 1\) ) on \((M, \hat{h})\) , which are slightly supercritical, under certain geometric and dimensional assumptions. These solutions certainly exhibit non-isolated blow-up. PubDate: 2017-10-13 DOI: 10.1007/s00526-017-1253-2 Issue No:Vol. 56, No. 6 (2017)

Authors:Miyuki Koiso; Bennett Palmer Abstract: Abstract We study the third and fourth variation of area for a compact domain in a constant mean curvature surface when there is a Killing field on \(\mathbf{R}^3\) whose normal component vanishes on the boundary. Examples are given to show that, in the presence of a zero eigenvalue, the non negativity of the second variation has no implications for the local area minimization of the surface. PubDate: 2017-10-10 DOI: 10.1007/s00526-017-1246-1 Issue No:Vol. 56, No. 6 (2017)

Authors:Mikhail Feldman; Adrian Tudorascu Abstract: Abstract We prove a weak-strong uniqueness result for the semi-geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre-Fenchel transform of the modified pressure. We give several examples where our results apply, including some classical solutions on the 2-dimensional torus, and the “stationary” solutions for 3DSG (for which the total wind velocity is zero but the pressure may be time-dependent). PubDate: 2017-10-09 DOI: 10.1007/s00526-017-1254-1 Issue No:Vol. 56, No. 6 (2017)

Authors:Cyril Imbert; Vinh Duc Nguyen Abstract: Abstract We are interested in the study of parabolic equations on a multi-dimensional junction, i.e. the union of a finite number of copies of a half-hyperplane of dimension \(d+1\) whose boundaries are identified. The common boundary is referred to as the junction hyperplane. The parabolic equations on the half-hyperplanes are in non-divergence form, fully non-linear and possibly degenerate, and they do degenerate and are quasi-convex along the junction hyperplane. More precisely, along the junction hyperplane the nonlinearities do not depend on second order derivatives and their sublevel sets with respect to the gradient variable are convex. The parabolic equations are supplemented with a non-linear boundary condition of Neumann type, referred to as a generalized junction condition, which is compatible with the maximum principle. Our main result asserts that imposing a generalized junction condition in a weak sense reduces to imposing an effective one in a strong sense. This result extends the one obtained by Imbert and Monneau for Hamilton–Jacobi equations on networks and multi-dimensional junctions. We give two applications of this result. On the one hand, we give the first complete answer to an open question about these equations: we prove in the two-domain case that the vanishing viscosity limit associated with quasi-convex Hamilton–Jacobi equations coincides with the maximal Ishii solution identified by Barles et al. (ESAIM Control Optim Calc Var 19(3):710–739, 2013). On the other hand, we give a short and simple PDE proof of a large deviation result of Boué et al. (Probab Theory Relat Fields 116:125–149, 2000). PubDate: 2017-10-07 DOI: 10.1007/s00526-017-1239-0 Issue No:Vol. 56, No. 6 (2017)

Authors:Jürgen Jost; Ruijun Wu; Miaomiao Zhu Abstract: Abstract The regularity of weak solutions of a two-dimensional nonlinear sigma model with coarse gravitino is shown. Here the gravitino is only assumed to be in \(L^p\) for some \(p>4\) . The precise regularity results depend on the value of p. PubDate: 2017-10-06 DOI: 10.1007/s00526-017-1241-6 Issue No:Vol. 56, No. 6 (2017)

Authors:Eleonora Cinti; Bruno Franchi; María del Mar González Abstract: Abstract Let \(\Omega \) be an open subset of a Stein manifold \(\Sigma \) and let M be its boundary. It is well known that M inherits a natural contact structure. In this paper we consider a family of variational functionals \(F_\varepsilon \) defined by the sum of two terms: a Dirichlet-type energy associated with a sub-Riemannian structure in \(\Omega \) and a potential term on the boundary M. We prove that the functionals \(F_\varepsilon \) \(\Gamma \) -converge to the intrinsic perimeter in M associated with its contact structure. Similar results have been obtained in the Euclidean space by Alberti, Bouchitté, Seppecher. We stress that already in the Euclidean setting the situation is not covered by the classical Modica–Mortola theorem because of the presence of the boundary term. We recall also that Modica–Mortola type results (without a boundary term) have been proved in the Euclidean space for sub-Riemannian energies by Monti and Serra Cassano. PubDate: 2017-10-06 DOI: 10.1007/s00526-017-1244-3 Issue No:Vol. 56, No. 6 (2017)

Authors:Leonid Berlyand; Etienne Sandier; Sylvia Serfaty Abstract: We propose an abstract framework for the homogenization of random functionals which may contain non-convex terms, based on a two-scale \(\Gamma \) -convergence approach and a definition of Young measures on micropatterns which encodes the profiles of the oscillating functions and of functionals. Our abstract result is a lower bound for such energies in terms of a cell problem (on large expanding cells) and the \(\Gamma \) -limits of the functionals at the microscale. We show that our method allows to retrieve the results of Dal Maso and Modica in the well-known case of the stochastic homogenization of convex Lagrangians. As an application, we also show how our method allows to stochastically homogenize a variational problem introduced and studied by Alberti and Müller, which is a paradigm of a problem where an additional mesoscale arises naturally due to the non-convexity of the singular perturbation (lower order) terms in the functional. PubDate: 2017-10-06 DOI: 10.1007/s00526-017-1249-y Issue No:Vol. 56, No. 6 (2017)

Authors:Flavia Giannetti; Antonia Passarelli di Napoli; Maria Alessandra Ragusa; Atsushi Tachikawa Abstract: Abstract We study the regularity of the local minimizers of non autonomous integral functionals of the type $$\begin{aligned} \int _\varOmega \varPhi ^{p(x)}\left( \big ( A^{\alpha \beta }_{ij}(x,u) D_iu^\alpha D_ju^\beta \big )^{1/2}\right) \, dx, \end{aligned}$$ where \(\varPhi \) is an Orlicz function satisfying both the \(\varDelta _2\) and the \(\nabla _2\) conditions, \(p(x):\varOmega \subset {{\mathbb {R}}}^{n}\rightarrow (1,+\infty )\) is continuous and the function \(A(x,s) = \big (A^{\alpha \beta }_{ij}(x,s)\big )\) is uniformly continuous. More precisely, under suitable assumptions on the functions \(\varPhi \) and p(x), we prove the Hölder continuity of the minimizers. Moreover, assuming in addition that the function \(A(x,s) = \big (A^{\alpha \beta }_{ij}(x,s)\big )\) is Hölder continuous, we prove the partial Hölder continuity of the gradient of the local minimizers too. PubDate: 2017-10-05 DOI: 10.1007/s00526-017-1248-z Issue No:Vol. 56, No. 6 (2017)

Authors:Yuanze Wu Abstract: Abstract Study the following K-component elliptic system Here \(k\ge 2\) is a integer and \(\Omega \subset \mathbb {R}^N(N\ge 4)\) is a bounded domain with smooth boundary \(\partial \Omega \) , \(a_i,\lambda _i>0\) , \(b_i\ge 0\) for all \(i=1,2,\ldots ,k\) and \(\beta <0\) , \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent. By the variational method, we obtain a nontrivial solution of this system. The concentration behavior of this nontrivial solution as \(\overrightarrow{\mathbf {b}}\rightarrow \overrightarrow{\mathbf {0}}\) and \(\beta \rightarrow -\infty \) are both studied and the phase separation is exhibited for \(N\ge 6\) , where \(\overrightarrow{\mathbf {b}}=(b_1,b_2,\ldots ,b_k)\) is a vector. Our results extend and generalize the results in Chen and Zou (Arch Ration Mech Anal 205:515–551, 2012; Calc Var Partial Differ Equ 52:423–467, 2015). Moreover, by studying the phase separation, we also prove some existence and multiplicity results of the sign-changing solutions to the following Brezís–Nirenberg problem of the Kirchhoff type $$\begin{aligned} \left\{ \begin{array}{ll} -\bigg (a+b\int _{\Omega } \nabla u ^2dx\bigg )\Delta u = \lambda u + u ^{2^*-2}u, &{}\quad \text {in }\Omega , \\ u =0,&{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ where \(N\ge 6\) , \(a,\lambda >0\) and \(b\ge 0\) . These results can be seen as an extension of the results in Cerami et al. (J Funct Anal 69:289–306, 1986). The concentration behaviors of the sign-changing solutions to the above equation as \(b\rightarrow 0^+\) are also obtained. PubDate: 2017-10-04 DOI: 10.1007/s00526-017-1252-3 Issue No:Vol. 56, No. 5 (2017)

Authors:Panu Lahti Abstract: Abstract In the setting of a metric space equipped with a doubling measure that supports a Poincaré inequality, we show that a set E is of finite perimeter if and only if \({\mathcal {H}}(\partial ^1 I_E)<\infty \) , that is, if and only if the codimension one Hausdorff measure of the 1-fine boundary of the set’s measure theoretic interior \(I_E\) is finite. To obtain the necessity of the above condition, we prove a suitable characterization of the 1-fine boundary, analogously to what is known in the case \(p>1\) , and apply a quasicontinuity-type result for \(\mathrm {BV}\) functions proved in the metric setting by Lahti and Shanmugalingam (J Math Pures Appl (9) 107(2):150–182, 2017). To obtain the sufficiency, we generalize further results of fine potential theory from the case \(p>1\) to the case \(p=1\) , including weak analogs of a Cartan property for solutions of obstacle problems, and of the Choquet property for finely open sets. PubDate: 2017-10-04 DOI: 10.1007/s00526-017-1242-5 Issue No:Vol. 56, No. 5 (2017)

Authors:Francesca Da Lio; Luca Martinazzi Abstract: Abstract In this paper we perform a blow-up and quantization analysis of the fractional Liouville equation in dimension 1. More precisely, given a sequence \(u_k :\mathbb {R}\rightarrow \mathbb {R}\) of solutions to 1 $$\begin{aligned} (-\Delta )^\frac{1}{2} u_k =K_ke^{u_k}\quad \text {in} \quad \mathbb {R}, \end{aligned}$$ 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 \(B=\{a_1,\ldots , a_N\}\) and that either (i) \(u_k\rightarrow u_\infty \) in \(W^{1,p}_{{{\mathrm{loc}}}}(\mathbb {R}{\setminus } B)\) and \(K_ke^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}K_\infty e^{u_\infty }+ \sum _{j=1}^N \pi \delta _{a_j}\) , or (ii) \(u_k\rightarrow -\infty \) uniformly locally in \(\mathbb {R}{\setminus } B\) and \(K_k e^{u_k} {\mathop {\rightharpoonup }\limits ^{*}}\sum _{j=1}^N \alpha _j \delta _{a_j}\) with \(\alpha _j\ge \pi \) for every j. This result, resting on the geometric interpretation and analysis of (1) provided in a recent collaboration of the authors with T. Rivière and on a classical work of Blank about immersions of the disk into the plane, is a fractional counterpart of the celebrated works of Brézis–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 allowed to change sign. PubDate: 2017-10-04 DOI: 10.1007/s00526-017-1245-2 Issue No:Vol. 56, No. 5 (2017)

Authors:Yuxiang Li; Lei Liu; Youde Wang Abstract: Abstract In this paper, we study the blow-up phenomena on the \(\alpha _k\) -harmonic map sequences with bounded uniformly \(\alpha _k\) -energy, denoted by \(\{u_{\alpha _k}: \alpha _k>1 \quad \text{ and } \quad \alpha _k\searrow 1\}\) , from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the \(\alpha _k\) -harmonic map sequence with respect to the corresponding \(\alpha _k\) -energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of \(\{u_{\alpha _k}\}\) as \(\alpha _k\searrow 1\) , the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence \(u_k:(\Sigma ,h_k)\rightarrow N\) , where the conformal class defined by \(h_k\) diverges, we also prove some similar results. PubDate: 2017-10-03 DOI: 10.1007/s00526-017-1211-z Issue No:Vol. 56, No. 5 (2017)

Authors:Mónica Clapp; Juan Carlos Fernández Abstract: Abstract Given a compact Riemannian manifold (M, g) without boundary of dimension \(m\ge 3\) and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation $$\begin{aligned} -\text {div}_{g}(a\nabla u)+bu=c u ^{2^{*}-2}u\quad \text { on }M, \end{aligned}$$ where \(a,b,c\in \mathcal {C}^{\infty }(M), a\) and c are positive, − div \(_{g}(a\nabla )+b\) is coercive, and \(2^{*}=\frac{2m}{m-2}\) is the critical Sobolev exponent. In particular, if \(R_{g}\) denotes the scalar curvature of (M, g), we give conditions which guarantee that the Yamabe problem $$\begin{aligned} \Delta _{g}u+\frac{m-2}{4(m-1)}R_{g}u=\kappa u^{2^{*}-2}\quad \text { on }M \end{aligned}$$ admits a prescribed number of nodal solutions. PubDate: 2017-10-03 DOI: 10.1007/s00526-017-1237-2 Issue No:Vol. 56, No. 5 (2017)

Authors:Tai-Chia Lin; Milivoj R. Belić; Milan S. Petrović; Hichem Hajaiej; Goong Chen Abstract: Abstract The virial theorem is a nice property for the linear Schrödinger equation in atomic and molecular physics as it gives an elegant ratio between the kinetic and potential energies and is useful in assessing the quality of numerically computed eigenvalues. If the governing equation is a nonlinear Schrödinger equation with power-law nonlinearity, then a similar ratio can be obtained but there seems to be no way of getting any eigenvalue estimates. It is surprising as far as we are concerned that when the nonlinearity is either square-root or saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue estimates of nonlinear Schrödinger (NLS) equations in \({{\mathbb {R}}^{2}}\) with square-root and saturable nonlinearity, respectively. Furthermore, we show here that the eigenvalue estimates can be used to obtain the 2nd order term (which is of order \(\ln \Gamma \) ) of the lower bound of the ground state energy as the coefficient \(\Gamma \) of the nonlinear term tends to infinity. PubDate: 2017-10-03 DOI: 10.1007/s00526-017-1251-4 Issue No:Vol. 56, No. 5 (2017)

Authors:Roberto Alicandro; Lucia De Luca; Adriana Garroni; Marcello Ponsiglione Abstract: Abstract In Alicandro et al. (J Mech Phys Solids 92:87–104, 2016) a simple discrete scheme for the motion of screw dislocations toward low energy configurations has been proposed. There, a formal limit of such a scheme, as the lattice spacing and the time step tend to zero, has been described. The limiting dynamics agrees with the maximal dissipation criterion introduced in Cermelli and Gurtin (Arch Ration Mech Anal 148, 1999) and predicts motion along the glide directions of the crystal. In this paper, we provide rigorous proofs of the results in [3], and in particular of the passage from the discrete to the continuous dynamics. The proofs are based on \(\Gamma \) -convergence techniques. PubDate: 2017-10-03 DOI: 10.1007/s00526-017-1247-0 Issue No:Vol. 56, No. 5 (2017)

Authors:Nassif Ghoussoub; Frédéric Robert Abstract: Abstract We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem \(L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{ x ^s}\) on a smooth bounded domain \(\Omega \) in \({\mathbb {R}}^n\) ( \(n\ge 3\) ) having the singularity 0 in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\) , \(0\le s <2\) , \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\le \lambda <\lambda _1(L_\gamma )\) , the latter being the first eigenvalue of the Hardy–Schrödinger operator \(L_\gamma :=-\Delta -\frac{\gamma }{ x ^2}\) . There is a threshold \(\lambda ^*(\gamma , \Omega ) \ge 0\) beyond which the minimal energy is achieved, but below which, it is not. It is well known that \(\lambda ^*(\Omega )=0\) in higher dimensions, for example if \(0\le \gamma \le \frac{(n-2)^2}{4}-1\) . Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when \( \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}\) , to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of \(\Omega \) and \(\gamma \) . If either \(s>0\) or if \(\gamma > 0\) , i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega \) , a notion that we introduce herein. On the other hand, and just like the case when \(\gamma =s=0\) studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), \(n=3\) is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when \(s=0\) , \(\gamma \le 0\) . PubDate: 2017-10-03 DOI: 10.1007/s00526-017-1238-1 Issue No:Vol. 56, No. 5 (2017)

Authors:Claudianor O. Alves; Marcos T. O. Pimenta Abstract: Abstract In this work we use variational methods to prove results on existence and concentration of solutions to a problem in \(\mathbb {R}^N\) involving the 1-Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation \(BV(\mathbb {R}^N)\) is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle due to Lions. PubDate: 2017-09-30 DOI: 10.1007/s00526-017-1236-3 Issue No:Vol. 56, No. 5 (2017)

Authors:C. A. Stuart Abstract: Abstract This paper concerns a functional of the form $$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$ on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \) . The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \) , but allow the Lagrangian to be singular at \(x=0\) . It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\) ), Legendre (ellipticity at \(u\equiv 0\) ) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \) ] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \) . Using recent criteria for the existence of a potential well of a \(C^1\) -functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases. PubDate: 2017-09-30 DOI: 10.1007/s00526-017-1250-5 Issue No:Vol. 56, No. 5 (2017)

Authors:Juncheng Wei; Matthias Winter; Wen Yang Abstract: Abstract We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable. PubDate: 2017-09-22 DOI: 10.1007/s00526-017-1233-6 Issue No:Vol. 56, No. 5 (2017)