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 Calculus of Variations and Partial Differential Equations   [SJR: 2.455]   [H-I: 44]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1432-0835 - ISSN (Online) 0944-2669    Published by Springer-Verlag  [2335 journals]
• Isoperimetry with upper mean curvature bounds and sharp stability
estimates
• Authors: B. Krummel; F. Maggi
Abstract: It was proved by Almgren that among boundaries whose mean curvature is bounded from above, perimeter is uniquely minimized by balls. We obtain sharp stability estimates for Almgren’s isoperimetric principle and, as an application, we deduce a sharp description of boundaries with almost constant mean curvature under a total perimeter bound which prevents bubbling.
PubDate: 2017-03-24
DOI: 10.1007/s00526-017-1139-3
Issue No: Vol. 56, No. 2 (2017)

• Infinitely many non-radial solutions for fractional Nirenberg problem
• Authors: Chungen Liu; Qiang Ren
Abstract: In this paper, the prescribed $$\sigma$$ -curvature problem \begin{aligned} P_{\sigma }^{g_0} u={\tilde{K}}(x)u^{\frac{N+2\sigma }{N-2\sigma }}, x\in {\mathbb {S}}^N,u>0 \end{aligned} is considered. When $${\tilde{K}}(x)$$ is some axis symmetric function on $${\mathbb {S}}^N$$ , by using singular perturbation method, it is proved that this problem possesses infinitely many non-radial solutions for $$0<\sigma \le 1$$ and $$N> 2\sigma +2$$ .
PubDate: 2017-03-23
DOI: 10.1007/s00526-017-1141-9
Issue No: Vol. 56, No. 2 (2017)

• Discreteness of interior transmission eigenvalues revisited
• Authors: Hoai-Minh Nguyen; Quoc-Hung Nguyen
Abstract: This paper is devoted to the discreteness of the transmission eigenvalue problems. It is known that this problem is not self-adjoint and a priori estimates are non-standard and do not hold in general. Two approaches are used. The first one is based on the multiplier technique and the second one is based on the Fourier analysis. The key point of the analysis is to establish the compactness and the uniqueness for Cauchy problems under various conditions. Using these approaches, we are able to rediscover quite a few known discreteness results in the literature and obtain various new results for which only the information near the boundary are required and there might be no contrast of the coefficients on the boundary.
PubDate: 2017-03-22
DOI: 10.1007/s00526-017-1143-7
Issue No: Vol. 56, No. 2 (2017)

• Regularity of $$\omega$$ ω -minimizers for a class of functionals with
non-standard growth
• Authors: Jihoon Ok
Abstract: We prove local Hölder continuity results for $$\omega$$ -minimizers of a class of functionals with non-standard growth, characterized by the fact of having a double type of degeneracy, and thereby extending to $$\omega$$ -minimizers the results obtained in Colombo and Mingione (Arch Ration Mech Anal 215(2):443–496, 2015) for ordinary minimizers. As a side benefit of the proof, we also consider a class of functionals with Orlicz-growth and prove regularity for $$\omega$$ -minimizers.
PubDate: 2017-03-21
DOI: 10.1007/s00526-017-1137-5
Issue No: Vol. 56, No. 2 (2017)

• On the p -Laplacian with Robin boundary conditions and boundary trace
theorems
• Authors: Hynek Kovařík; Konstantin Pankrashkin
Abstract: Let $$\Omega \subset \mathbb {R}^\nu$$ , $$\nu \ge 2$$ , be a $$C^{1,1}$$ domain whose boundary $$\partial \Omega$$ is either compact or behaves suitably at infinity. For $$p\in (1,\infty )$$ and $$\alpha >0$$ , define \begin{aligned} \Lambda (\Omega ,p,\alpha ):=\inf _{\begin{array}{c} u\in W^{1,p}(\Omega )\\ u\not \equiv 0 \end{array}}\dfrac{\displaystyle \int _\Omega \nabla u ^p \mathrm {d} x - \alpha \displaystyle \int _{\partial \Omega } u ^p\mathrm {d}\sigma }{\displaystyle \int _\Omega u ^p\mathrm {d} x}, \end{aligned} where $$\mathrm {d}\sigma$$ is the surface measure on $$\partial \Omega$$ . We show the asymptotics \begin{aligned} \Lambda (\Omega ,p,\alpha )=-(p-1)\alpha ^{\frac{p}{p-1}} - (\nu -1)H_\mathrm {max}\, \alpha + o(\alpha ), \quad \alpha \rightarrow +\infty , \end{aligned} where $$H_\mathrm {max}$$ is the maximum mean curvature of $$\partial \Omega$$ . The asymptotic behavior of the associated minimizers is discussed as well. The estimate is then applied to the study of the best constant in a boundary trace theorem for expanding domains, to the norm estimate for extension operators and to related isoperimetric inequalities.
PubDate: 2017-03-21
DOI: 10.1007/s00526-017-1138-4
Issue No: Vol. 56, No. 2 (2017)

• Estimates for the first eigenvalue of Jacobi operator on hypersurfaces
with constant mean curvature in spheres
• Authors: Daguang Chen; Qing-Ming Cheng
Abstract: In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere $$S^{n+1}(1)$$ . We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, $$\varphi :\ M \rightarrow S^{n+1}(1)$$ is isometric to $$S^1(r)\times S^{n-1}(\sqrt{1-r^2})$$ when $$H\ne 0$$ or $$\varphi :\ M \rightarrow S^{n+1}(1)$$ is isometric to a Clifford torus $$S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right)$$ , for $$k=1, 2, \ldots , n-1$$ when $$H=0$$ .
PubDate: 2017-03-21
DOI: 10.1007/s00526-017-1132-x
Issue No: Vol. 56, No. 2 (2017)

• Global Lorentz estimates for nonlinear parabolic equations on nonsmooth
domains
• Authors: The Anh Bui; Xuan Thinh Duong
Abstract: Consider the nonlinear parabolic equation in the form \begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,( F ^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned} where $$T>0$$ and $$\Omega$$ is a Reifenberg domain. We suppose that the nonlinearity $${\mathbf {a}}(\xi ,x,t)$$ has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity $${\mathbf {a}}(\xi ,x,t)$$ and to more general setting of Lorentz spaces.
PubDate: 2017-03-20
DOI: 10.1007/s00526-017-1130-z
Issue No: Vol. 56, No. 2 (2017)

• Stability of the traveling waves for the derivative Schrödinger
equation in the energy space
• Authors: Changxing Miao; Xingdong Tang; Guixiang Xu
Abstract: In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel–Merle–Tsai’s analytic approach in Martel et al. (Commun Math Phys 231(2):347–373, 2002, Duke Math J 133(3):405–466, 2006). As a by-product, we also give an alternative proof of the stability of the single traveling wave in the energy space in Colin and Ohta (Ann Inst Henri Poincaré Anal Non Linéaire 23(5):753–764, 2006), where Colin and Ohta made use of the concentration-compactness argument.
PubDate: 2017-03-18
DOI: 10.1007/s00526-017-1128-6
Issue No: Vol. 56, No. 2 (2017)

• Global gradient estimates for non-uniformly elliptic equations
• Authors: Sun-Sig Byun; Jehan Oh
Abstract: We consider a nonlinear and non-uniformly elliptic problem in divergence form on a bounded domain. The problem under consideration is characterized by the fact that its ellipticity rate and growth radically change with the position, which provides a model for describing a feature of strongly anisotropic materials. We establish the global Calderón–Zygmund type estimates for the distributional solution in the case that the boundary of the domain is of class $$C^{1,\beta }$$ for some $$\beta >0$$ .
PubDate: 2017-03-18
DOI: 10.1007/s00526-017-1148-2
Issue No: Vol. 56, No. 2 (2017)

• Semilinear elliptic equations with Hardy potential and subcritical source
term
• Authors: Phuoc-Tai Nguyen
Abstract: Let $$\Omega$$ be a smooth bounded domain in $${\mathbb {R}}^N$$ ( $$N>2$$ ) and $$\delta (x):=\text {dist}\,(x,\partial \Omega )$$ . Assume $$\mu \in {\mathbb {R}}_+, \nu$$ is a nonnegative finite measure on $$\partial \Omega$$ and $$g \in C(\Omega \times {\mathbb {R}}_+)$$ . We study positive solutions of P \begin{aligned} -\Delta u - \frac{\mu }{\delta ^2} u = g(x,u) \text { in } \Omega , \qquad \text {tr}^*(u)=\nu . \end{aligned} Here $$\text {tr}^*(u)$$ denotes the normalized boundary trace of u which was recently introduced by Marcus and Nguyen (Ann Inst H Poincaré Anal Non Linéaire, 34, 69–88, 2017). We focus on the case $$0<\mu < C_H(\Omega )$$ (the Hardy constant for $$\Omega$$ ) and provide qualitative properties of positive solutions of (P). When $$g(x,u)=u^q$$ with $$q>0$$ , we prove that there is a critical value $$q^*$$ (depending only on $$N, \mu$$ ) for (P) in the sense that if $$q<q^*$$ then (P) possesses a solution under a smallness assumption on $$\nu$$ , but if $$q \ge q^*$$ this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where g is subcritical [see (1.28)]. We also investigate the case where g is linear or sublinear and give an existence result for (P).
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1144-6
Issue No: Vol. 56, No. 2 (2017)

• On minimizers for the isotropic–nematic interface problem
• Authors: Jinhae Park; Wei Wang; Pingwen Zhang; Zhifei Zhang
Abstract: In this paper, we investigate the structure and stability of the isotropic-nematic interface in 1-D. In the absence of the anisotropic energy, the uniaxial solution is the only global minimizer. In the presence of the anisotropic energy, the uniaxial solution with the homeotropic anchoring is stable for $$L_2<0$$ and unstable for $$L_2>0$$ . We also present many interesting open questions, some of which are related to De Giorgi conjecture.
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1131-y
Issue No: Vol. 56, No. 2 (2017)

• Global solutions of nonlinear Schrödinger equations
• Authors: Martin Schechter
Abstract: We study the nonlinear Schrödinger equation in $$\mathbb {R}^n$$ without making any periodicity assumptions on the potential or on the nonlinear term. This prevents us from using concentration compactness methods. Our assumptions are such that the potential does not change the essential spectrum of the linear operator. This results in $$[0, \infty )$$ being the absolutely continuous part of the spectrum. If there are an infinite number of negative eigenvalues, they will converge to 0. In each case we obtain nontrivial solutions. We also obtain least energy solutions.
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1145-5
Issue No: Vol. 56, No. 2 (2017)

• Monotonicity of solutions for some nonlocal elliptic problems in
half-spaces
• Authors: B. Barrios; L. Del Pezzo; J. García-Melián; A. Quaas
Abstract: In this paper we consider classical solutions u of the semilinear fractional problem $$(-\Delta )^s u = f(u)$$ in $${\mathbb {R}}^N_+$$ with $$u=0$$ in $${\mathbb {R}}^N {\setminus } {\mathbb {R}}^N_+$$ , where $$(-\Delta )^s$$ , $$0<s<1$$ , stands for the fractional laplacian, $$N\ge 2$$ , $${\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N{:}\ x_N>0\}$$ is the half-space and $$f\in C^1$$ is a given function. With no additional restriction on the function f, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in $${\mathbb {R}}^N_+$$ and verify \begin{aligned} \frac{\partial u}{\partial x_N}>0 \quad \hbox {in } {\mathbb {R}}^N_+. \end{aligned} This is in contrast with previously known results for the local case $$s=1$$ , where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when $$f(0)<0$$ .
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1133-9
Issue No: Vol. 56, No. 2 (2017)

• On the effect of interactions beyond nearest neighbours on non-convex
lattice systems
• Authors: Roberto Alicandro; Giuliano Lazzaroni; Mariapia Palombaro
Abstract: We analyse the rigidity of non-convex discrete energies where at least nearest and next-to-nearest neighbour interactions are taken into account. Our purpose is to show that interactions beyond nearest neighbours have the role of penalising changes of orientation and, to some extent, they may replace the positive-determinant constraint that is usually required when only nearest neighbours are accounted for. In a discrete to continuum setting, we prove a compactness result for a family of surface-scaled energies and we give bounds on its possible Gamma-limit in terms of interfacial energies that penalise changes of orientation.
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1129-5
Issue No: Vol. 56, No. 2 (2017)

• Analytic aspects of the Tzitzéica equation: blow-up analysis and
existence results
• Authors: Aleks Jevnikar; Wen Yang
Abstract: We are concerned with the following class of equations with exponential nonlinearities: \begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned} which is related to the Tzitzéica equation. Here $$h_1, h_2$$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitzéica equation on compact surfaces: we start by proving a sharp Moser–Trudinger inequality related to this problem. Finally, we give a general existence result.
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1136-6
Issue No: Vol. 56, No. 2 (2017)

• Solitary waves for coupled nonlinear elliptic system with nonhomogeneous
nonlinearities
• Authors: Jun Wang
Abstract: In present paper we consider a class of coupled elliptic system with nonhomogeneous nonlinearities. This type of system is related to the Raman amplification in a plasma. We make rigorous study and find the threshold conditions to guarantee the existence, nonexistence and multiplicity of nontrivial solutions for both two and three coupled system by using Morse theory, direct analysis methods and Krasnosel’skii–Rabinowitz global bifurcation theorem. Moreover, we study the asymptotical behavior of positive solutions, and prove some interesting phenomena for these solutions. Comparing to our previous works Wang and Shi (standing waves for weakly coupled Schrödinger equations with quadratic nonlinearities. Preprint, 2015) on the homogeneous case, we encounter some new challenges in proving the existence and multiplicity of nontrivial solutions. We overcome these difficult by combining the Mountain–Pass theorem in convex set and the Nehari constraint methods.
PubDate: 2017-03-17
DOI: 10.1007/s00526-017-1147-3
Issue No: Vol. 56, No. 2 (2017)

• Generalized Ginzburg–Landau equations in high dimensions
• Authors: Yuxin Ge; Etienne Sandier; Peng Zhang
Abstract: In this work, we study critical points of the generalized Ginzburg–Landau equations in dimensions $$n\ge 3$$ which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equations tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres, and the convergence is strong away from a finite set consisting (1) of the infinite energy singularities of the limiting map, and (2) of points where bubbling off of finite energy n-harmonic maps could take place. The latter case is specific to dimensions $${>}2$$ . We also exhibit a criticality condition satisfied by the limiting n-harmonic maps which constrains the location of the infinite energy singularities. Finally we construct an example of non-minimizing solutions to the generalized Ginzburg–Landau equations satisfying our assumptions.
PubDate: 2017-03-16
DOI: 10.1007/s00526-017-1134-8
Issue No: Vol. 56, No. 2 (2017)

• On the subRiemannian cut locus in a model of free two-step Carnot group
• Authors: Annamaria Montanari; Daniele Morbidelli
Abstract: We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.
PubDate: 2017-03-16
DOI: 10.1007/s00526-017-1149-1
Issue No: Vol. 56, No. 2 (2017)

• Fractional Poincaré inequality with finite total Q -curvature
• Authors: Yannick Sire; Yi Wang
Abstract: In this paper, we prove several Poincaré inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the Q-curvature on noncompact complete manifolds.
PubDate: 2017-03-15
DOI: 10.1007/s00526-017-1146-4
Issue No: Vol. 56, No. 2 (2017)

• An eigenvalue problem for a fully nonlinear elliptic equation with
• Authors: Ryan Hynd
Abstract: We consider the problem of finding $$\lambda \in {\mathbb {R}}$$ and a function $$u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$$ that satisfy the PDE \begin{aligned} \max \left\{ \lambda + F(D^2u) -f(x),H(Du)\right\} =0, \quad x\in {\mathbb {R}}^n. \end{aligned} Here F is elliptic, positively homogeneous and superadditive, f is convex and superlinear, and H is typically assumed to be convex. Examples of this type of PDE arise in the theory of singular ergodic control. We show that there is a unique $$\lambda ^*$$ for which the above equation has a solution u with appropriate growth as $$x \rightarrow \infty$$ . Moreover, associated to $$\lambda ^*$$ is a convex solution $$u^*$$ that has essentially bounded second derivatives, provided F is uniformly elliptic and H is uniformly convex. It is unknown whether or not $$u^*$$ is unique up to an additive constant; however, we verify that this is the case when $$n=1$$ or when F, f, H are “rotational.”
PubDate: 2017-03-09
DOI: 10.1007/s00526-017-1115-y
Issue No: Vol. 56, No. 2 (2017)

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