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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  [2353 journals]
• On Neumann problems for nonlocal Hamilton–Jacobi equations with
• Authors: Daria Ghilli
Abstract: Abstract We are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton–Jacobi equations related to jump processes in general smooth domains. We consider a nonlocal diffusive term of censored type of order strictly less than 1 and Hamiltonians both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron’s method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data.
PubDate: 2017-09-01
DOI: 10.1007/s00526-017-1225-6
Issue No: Vol. 56, No. 5 (2017)

• A quantitative form of Faber–Krahn inequality
• Authors: Nicola Fusco; Yi Ru-Ya Zhang
Abstract: Abstract The Faber–Krahn inequality states that balls are the unique minimizers of the first eigenvalue of the p-Laplacian among all sets with fixed volume. In this paper we prove a sharp quantitative form of this inequality. This extends to the case $$p>1$$ a recent result proved by Brasco et al. (Duke Math J 164:1777–1831, 2015) for the Laplacian.
PubDate: 2017-08-30
DOI: 10.1007/s00526-017-1224-7
Issue No: Vol. 56, No. 5 (2017)

• Optimal lower exponent for the higher gradient integrability of solutions
to two-phase elliptic equations in two dimensions
• Authors: Silvio Fanzon; Mariapia Palombaro
Abstract: Abstract We study the higher gradient integrability of distributional solutions u to the equation $${{\mathrm{div}}}(\sigma \nabla u) = 0$$ in dimension two, in the case when the essential range of $$\sigma$$ consists of only two elliptic matrices, i.e., $$\sigma \in \{\sigma _1, \sigma _2\}$$ a.e. in $$\Omega$$ . In Nesi et al. (Ann Inst H Poincaré Anal Non Linéaire 31(3):615–638, 2014), for every pair of elliptic matrices $$\sigma _1$$ and $$\sigma _2$$ , exponents $$p_{\sigma _1,\sigma _2}\in (2,+\infty )$$ and $$q_{\sigma _1,\sigma _2}\in (1,2)$$ have been found so that if $$u\in W^{1,q_{\sigma _1,\sigma _2}}(\Omega )$$ is solution to the elliptic equation then $$\nabla u\in L^{p_{\sigma _1,\sigma _2}}_{\mathrm{weak}}(\Omega )$$ and the optimality of the upper exponent $$p_{\sigma _1,\sigma _2}$$ has been proved. In this paper we complement the above result by proving the optimality of the lower exponent $$q_{\sigma _1,\sigma _2}$$ . Precisely, we show that for every arbitrarily small $$\delta$$ , one can find a particular microgeometry, i.e., an arrangement of the sets $$\sigma ^{-1}(\sigma _1)$$ and $$\sigma ^{-1}(\sigma _2)$$ , for which there exists a solution u to the corresponding elliptic equation such that $$\nabla u \in L^{q_{\sigma _1,\sigma _2}-\delta }$$ , but $$\nabla u \notin L^{q_{\sigma _1,\sigma _2}}$$ . The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in Astala et al. (Ann Scuola Norm Sup Pisa Cl Sci 5(7):1–50, 2008) for the isotropic case.
PubDate: 2017-08-29
DOI: 10.1007/s00526-017-1222-9
Issue No: Vol. 56, No. 5 (2017)

• Collision of incompressible inviscid fluids effluxing from two nozzles
• Authors: Lili Du; Yongfu Wang
Abstract: Abstract This paper concerns the mathematical theory of the collision problem of two-dimensional incompressible inviscid fluids issuing from two given nozzles. The main result reads that for given two co-axis symmetric semi-infinitely long nozzles with arbitrary variable sections, imposing the incoming mass fluxes in two nozzles, there exists a smooth impinging outgoing jet, such that the two free boundaries of the impinging jet initiate smoothly at the endpoints of the nozzles and approach to some asymptotic direction in downstream, and the pressure on the free surface remains a constant. Furthermore, we show that there exists a unique smooth surface separating the two nonmiscible fluids and there exists a unique stagnation point in the fluid region and its closure. Moreover, some results on the uniqueness and the estimates of the location of the impinging outgoing jet are also established. Finally, the asymptotic behaviors, the precise estimate to the deflection angle and other properties to the impinging outgoing jet are also considered.
PubDate: 2017-08-28
DOI: 10.1007/s00526-017-1221-x
Issue No: Vol. 56, No. 5 (2017)

• Existence and multiplicity of positive solutions to Schrödinger–Poisson
type systems with critical nonlocal term
• Authors: Yuhua Li; Fuyi Li; Junping Shi
Abstract: Abstract The existence, nonexistence and multiplicity of positive radially symmetric solutions to a class of Schrödinger–Poisson type systems with critical nonlocal term are studied with variational methods. The existence of both the ground state solution and mountain pass type solutions are proved. It is shown that the parameter ranges of existence and nonexistence of positive solutions for the critical nonlocal case are completely different from the ones for the subcritical nonlocal system.
PubDate: 2017-08-24
DOI: 10.1007/s00526-017-1229-2
Issue No: Vol. 56, No. 5 (2017)

• Uniform Lipschitz regularity of flat segregated interfaces in a singularly
perturbed problem
• Authors: Kelei Wang
Abstract: Abstract For the singularly perturbed system \begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned} we prove that flat segregated interfaces are uniformly Lipschitz as $$\beta \rightarrow +\infty$$ . As a byproduct of the proof we also obtain the optimal lower bound near flat interfaces, \begin{aligned} \sum _iu_{i,\beta }\ge c\beta ^{-1/4}. \end{aligned} .
PubDate: 2017-08-24
DOI: 10.1007/s00526-017-1235-4
Issue No: Vol. 56, No. 5 (2017)

• Universal inequality and upper bounds of eigenvalues for non-integer
poly-Laplacian on a bounded domain
• Authors: Hua Chen; Ao Zeng
Abstract: Abstract In this paper we first study the universal inequality which is related to the eigenvalues of the fractional Laplacian $$(-\Delta )^s _{\Omega }$$ for $$s>0$$ and $$s\in \mathbb {Q}_+$$ . Here $$\Omega \subset \mathbb {R}^n$$ is a bounded open domain, and $$\mathbb {Q}_+$$ is the set of all positive rational numbers. Secondly, if $$s\in \mathbb {Q}_+$$ and $$s\ge 1$$ (in this case, the operator is also called the non-integer poly-Laplacian), then by this universal inequality and the variant of Chebyshev sum inequality, we can deduce the so-called Yang type inequality for the corresponding eigenvalue problem, which is the extension to the case of poly-Laplacian operators. Finally, we can get the upper bounds of the corresponding eigenvalues from the Yang type inequality.
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1220-y
Issue No: Vol. 56, No. 5 (2017)

• Global dynamics of the Lotka–Volterra competition–diffusion system
with equal amount of total resources, III
• Authors: Xiaoqing He; Wei-Ming Ni
Abstract: Abstract In this paper—Part III of this series of three papers, we continue to investigate the joint effects of diffusion and spatial concentration on the global dynamics of the classical Lotka–Volterra competition–diffusion system. To further illustrate the general results obtained in Part I (He and Ni in Commun Pure Appl Math 69:981–1014, 2016. doi:10.1002/cpa.21596), we have focused on the case when the two competing species have identical competition abilities and the same amount of total resources. In contrast to Part II (He and Ni in Calc Var Partial Differ Equ 2016. doi:10.1007/s00526-016-0964-0), our results here show that in case both species have spatially heterogeneous distributions of resources, the outcome of the competition is independent of initial values but depends solely on the dispersal rates, which in turn depends on the distribution profiles of the resources—thereby extending the celebrated phenomenon “slower diffuser always prevails!” Furthermore, the species with a “sharper” spatial concentration in its distribution of resources seems to have the edge of competition advantage. Limiting behaviors of the globally asymptotically stable steady states are also obtained under various circumstances in terms of dispersal rates.
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1234-5
Issue No: Vol. 56, No. 5 (2017)

• Defects of liquid crystals with variable degree of orientation
• Authors: Onur Alper; Robert Hardt; Fang-Hua Lin
Abstract: Abstract The defect set of minimizers of the modified Ericksen energy for nematic liquid crystals consists locally of a finite union of isolated points and Hölder continuous curves with finitely many crossings.
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1218-5
Issue No: Vol. 56, No. 5 (2017)

• Normalized bound states for the nonlinear Schrödinger equation in
bounded domains
• Authors: Dario Pierotti; Gianmaria Verzini
Abstract: Abstract Given $$\rho >0$$ , we study the elliptic problem \begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U= U ^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned} where $$\Omega \subset {\mathbb {R}}^N$$ is a bounded domain and $$p>1$$ is Sobolev-subcritical, searching for conditions (about $$\rho$$ , N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is $$L^2$$ -subcritical, i.e. $$1<p<1+4/N$$ , the problem admits solutions for every $$\rho >0$$ . In the $$L^2$$ -critical and supercritical case, i.e. when $$1+4/N \le p < 2^*-1$$ , we show that, for any $$k\in {\mathbb {N}}$$ , the problem admits solutions having Morse index bounded above by k only if $$\rho$$ is sufficiently small. Next we provide existence results for certain ranges of $$\rho$$ , which can be estimated in terms of the Dirichlet eigenvalues of $$-\Delta$$ in $$H^1_0(\Omega )$$ , extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1232-7
Issue No: Vol. 56, No. 5 (2017)

• Optimal regularity for the thin obstacle problem with $$C^{0,\alpha }$$ C
0 , α coefficients
• Authors: Angkana Rüland; Wenhui Shi
Abstract: Abstract In this article we study solutions to the (interior) thin obstacle problem under low regularity assumptions on the coefficients, the obstacle and the underlying manifold. Combining the linearization method of Andersson (Invent Math 204(1):1–82, 2016. doi:10.1007/s00222-015-0608-6) and the epiperimetric inequality from Focardi and Spadaro (Adv Differ Equ 21(1–2):153–200, 2016), Garofalo, Petrosyan and Smit Vega Garcia (J Math Pures Appl 105(6):745–787, 2016. doi:10.1016/j.matpur.2015.11.013), we prove the optimal $$C^{1,\min \{\alpha ,1/2\}}$$ regularity of solutions in the presence of $$C^{0,\alpha }$$ coefficients $$a^{ij}$$ and $$C^{1,\alpha }$$ obstacles $$\phi$$ . Moreover we investigate the regularity of the regular free boundary and show that it has the structure of a $$C^{1,\gamma }$$ manifold for some $$\gamma \in (0,1)$$ .
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1230-9
Issue No: Vol. 56, No. 5 (2017)

• Decay of superconductivity away from the magnetic zero set
• Authors: Bernard Helffer; Ayman Kachmar
Abstract: Abstract We establish exponential bounds on the Ginzburg–Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg–Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is Hölder continuous.
PubDate: 2017-08-23
DOI: 10.1007/s00526-017-1226-5
Issue No: Vol. 56, No. 5 (2017)

• On the interior regularity of weak solutions to the 2-D incompressible
Euler equations
• Authors: Juhana Siljander; José Miguel Urbano
Abstract: Abstract We study whether some of the non-physical properties observed for weak solutions of the incompressible Euler equations can be ruled out by studying the vorticity formulation. Our main contribution is in developing an interior regularity method in the spirit of De Giorgi–Nash–Moser, showing that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity result \begin{aligned} u \in L_\mathrm{loc}^{2+\varepsilon }(\Omega _T) \implies \mathrm{local\ regularity} \end{aligned} for weak solutions in the energy space $$L_t^\infty L_x^2$$ , satisfying appropriate vorticity estimates. We also obtain improved integrability for the vorticity—which is to be compared with the DiPerna–Lions assumptions. The argument is completely local in nature as the result follows from the structural properties of the equation alone, while completely avoiding all sorts of boundary conditions and related gradient estimates. To the best of our knowledge, the approach we follow is new in the context of Euler equations and provides an alternative look at interior regularity issues. We also show how our method can be used to give a modified proof of the classical Serrin condition for the regularity of the Navier–Stokes equations in any dimension.
PubDate: 2017-08-21
DOI: 10.1007/s00526-017-1231-8
Issue No: Vol. 56, No. 5 (2017)

• Interior Sobolev regularity for fully nonlinear parabolic equations
• Authors: Ricardo Castillo; Edgard A. Pimentel
Abstract: Abstract In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in $$W^{2,1;p}_{ loc }$$ . Our argument unfolds by importing improved regularity from a limiting configuration. In this concrete case, we recur to the recession function associated with F. This machinery allows us to impose conditions solely on the original operator at the infinity of $$\mathcal {S}(d)$$ . From a heuristic viewpoint, integral regularity would be set by the behavior of F at the ends of that space. Moreover, we explore a number of consequences of our findings, and develop some related results; these include a parabolic version of Escauriaza’s exponent, a universal modulus of continuity for the solutions and estimates in p-BMO spaces.
PubDate: 2017-08-21
DOI: 10.1007/s00526-017-1227-4
Issue No: Vol. 56, No. 5 (2017)

• Generalized characteristics and Lax–Oleinik operators: global theory
• Authors: Piermarco Cannarsa; Wei Cheng
Abstract: Abstract For autonomous Tonelli systems on $$\mathbb {R}^n$$ , we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.
PubDate: 2017-08-21
DOI: 10.1007/s00526-017-1219-4
Issue No: Vol. 56, No. 5 (2017)

• Global properties of Dirichlet forms in terms of Green’s formula
• Authors: Sebastian Haeseler; Matthias Keller; Daniel Lenz; Jun Masamune; Marcel Schmidt
Abstract: Abstract We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green’s formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first present results in the framework of general Dirichlet forms on $$\sigma$$ -finite measure spaces. For regular Dirichlet forms our results can be strengthened as all operators from the previous considerations turn out to be restrictions of a single operator. Finally, the results are applied to graphs, weighted manifolds, and metric graphs, where the operators under investigation can be determined rather explicitly, and certain volume growth criteria can be (re)derived.
PubDate: 2017-08-05
DOI: 10.1007/s00526-017-1216-7
Issue No: Vol. 56, No. 5 (2017)

• General methods of elliptic minimization
• Authors: J. Harrison; H. Pugh
Abstract: Abstract We provide general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. Given a collection of competing “surfaces” that span a given “bounding set” in an ambient metric space, we produce one minimizing an elliptic area functional. The collection of competing surfaces is assumed to satisfy a set of geometrically-defined axioms. These axioms hold for collections defined using any combination of homological, cohomological or linking number spanning conditions. A variety of minimization problems can be solved, including sliding boundaries.
PubDate: 2017-08-03
DOI: 10.1007/s00526-017-1217-6
Issue No: Vol. 56, No. 4 (2017)

• A multidimensional Birkhoff theorem for time-dependent Tonelli
Hamiltonians
• Authors: Marie-Claude Arnaud; Andrea Venturelli
Abstract: Abstract Let M be a closed and connected manifold, $$H:T^*M\times {{\mathbb {R}}}/\mathbb {Z}\rightarrow \mathbb {R}$$ a Tonelli 1-periodic Hamiltonian and $${\mathscr {L}}\subset T^*M$$ a Lagrangian submanifold Hamiltonianly isotopic to the zero section. We prove that if $${\mathscr {L}}$$ is invariant by the time-one map of H, then $${\mathscr {L}}$$ is a graph over M. An interesting consequence in the autonomous case is that in this case, $${\mathscr {L}}$$ is invariant by all the time t maps of the Hamiltonian flow of H.
PubDate: 2017-07-21
DOI: 10.1007/s00526-017-1210-0
Issue No: Vol. 56, No. 4 (2017)

• Some remarks on the classification of global solutions with asymptotically
flat level sets
• Abstract: Abstract We simplify some technical steps from Savin (Ann Math. (2) 169(1):41–78, 2009) in which a conjecture of De Giorgi was addressed. For completeness we make the paper self-contained and reprove the classification of certain global bounded solutions for semilinear equations of the type \begin{aligned} \triangle u=W^{\prime }(u), \end{aligned} where W is a double well potential.
PubDate: 2017-09-08
DOI: 10.1007/s00526-017-1228-3

• A chromaticity-brightness model for color images denoising in a Meyer’s
“u + v” framework
• Abstract: Abstract A variational model for imaging segmentation and denoising color images is proposed. The model combines Meyer’s “u+v” decomposition with a chromaticity-brightness framework and is expressed by a minimization of energy integral functionals depending on a small parameter $$\varepsilon >0$$ . The asymptotic behavior as $$\varepsilon \rightarrow 0^+$$ is characterized, and convergence of infima, almost minimizers, and energies are established. In particular, an integral representation of the lower semicontinuous envelope, with respect to the $$L^1$$ -norm, of functionals with linear growth and defined for maps taking values on a certain compact manifold is provided. This study escapes the realm of previous results since the underlying manifold has boundary, and the integrand and its recession function fail to satisfy hypotheses commonly assumed in the literature. The main tools are $$\Gamma$$ -convergence and relaxation techniques.
PubDate: 2017-09-07
DOI: 10.1007/s00526-017-1223-8

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