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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  [2355 journals]
• Fractional Sobolev metrics on spaces of immersed curves
• Authors: Martin Bauer; Martins Bruveris; Boris Kolev
Abstract: Abstract Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves $$\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$$ and on its Sobolev completions $${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$ . We prove local well-posedness of the geodesic equations both on the Banach manifold $${\mathcal {I}}^{q}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$ and on the Fréchet-manifold $$\mathrm {Imm}(\mathrm {S}^{1},\mathbb {R}^d)$$ provided the order of the metric is greater or equal to one. In addition we show that the $$H^s$$ -metric induces a strong Riemannian metric on the Banach manifold $${\mathcal {I}}^{s}(\mathrm {S}^{1},{\mathbb {R}}^{d})$$ of the same order s, provided $$s>\frac{3}{2}$$ . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group.
PubDate: 2018-01-13
DOI: 10.1007/s00526-018-1300-7
Issue No: Vol. 57, No. 1 (2018)

• Polyconvexity for functions of a system of closed differential forms
• Authors: M. Šilhavý
Abstract: Abstract This paper deals with the weakened convexity properties, mult. ext. quasiconvexity, mult. ext. one convexity, and mult. ext. polyconvexity, for integral functionals of the form \begin{aligned} I(\omega _1,\ldots ,\omega _s) = \int _\Omega f(\omega _1,\ldots ,\omega _s) dx \end{aligned} where $$\omega _1,\ldots ,\omega _s$$ are closed differential forms on a bounded open set $$\Omega \subset {\mathbb {R}}^n$$ . The main results of the paper are explicit descriptions of mult. ext. quasiaffine and mult ext. polyconvex functions. It turns out that these two classes consist, respectively, of linear and convex combinations of the set of all wedge products of exterior powers of the forms $$\omega _1,\ldots ,\omega _s$$ . Thus, for example, a function $$f=f(\omega _1,\ldots ,\omega _s)$$ is mult. ext. polyconvex if and only if \begin{aligned} f(\omega _1,\ldots ,\omega _s) = \Phi (\ldots ,\omega _1^{q_1}\wedge \cdots \wedge \omega _s^{q_s},\ldots ) \end{aligned} where $$q_1,\ldots ,q_s$$ ranges a finite set of integers and $$\Phi$$ is a convex function. An existence theorem for the minimum energy state is proved for mult. ext. polyconvex integrals. The polyconvexity in the calculus of variations and nonlinear elasticity are particular cases of mult. ext. polyconvexity. Our main motivation comes from electro-magneto-elastic interactions in continuous bodies. There the mult. ext. polyconvexity takes the form determined by an involved direct calculation in an earlier paper of the author (Šilhavý in Math Mech Solids, 2017. http://journals.sagepub.com/doi/metrics/10.1177/1081286517696536).
PubDate: 2018-01-10
DOI: 10.1007/s00526-017-1298-2
Issue No: Vol. 57, No. 1 (2018)

• Neumann heat flow and gradient flow for the entropy on non-convex domains
• Authors: Janna Lierl; Karl-Theodor Sturm
Abstract: Abstract For large classes of non-convex subsets Y in $$\mathbb R^n$$ or in Riemannian manifolds (M, g) or in RCD-spaces (X, d, m) we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $$(Y,d_Y,m_Y)$$ exists—despite the fact that the entropy is not semiconvex—and coincides with the heat flow on Y with Neumann boundary conditions.
PubDate: 2018-01-06
DOI: 10.1007/s00526-017-1292-8
Issue No: Vol. 57, No. 1 (2018)

• Discrete minimisers are close to continuum minimisers for the interaction
energy
• Authors: J. A. Cañizo; F. S. Patacchini
Abstract: Abstract Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $$\Gamma$$ -converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural iscrete analogue containing all the discrete minimisers.
PubDate: 2018-01-02
DOI: 10.1007/s00526-017-1289-3
Issue No: Vol. 57, No. 1 (2018)

• Authors: Lucas Ambrozio; Alessandro Carlotto; Ben Sharp
Abstract: Abstract We investigate compactness phenomena involving free boundary minimal hypersurfaces in Riemannian manifolds of dimension less than eight. We provide natural geometric conditions that ensure strong one-sheeted graphical subsequential convergence, discuss the limit behaviour when multi-sheeted convergence happens and derive various consequences in terms of finiteness and topological control.
PubDate: 2017-12-26
DOI: 10.1007/s00526-017-1281-y
Issue No: Vol. 57, No. 1 (2017)

• Bifurcations from the orbit of solutions of the Neumann problem
• Authors: Anna Gołȩbiewska; Joanna Kluczenko; Piotr Stefaniak
Abstract: Abstract The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient conditions for local and global bifurcations, in terms of the right-hand side of the system and eigenvalues of the Laplace operator. Moreover, we characterise orbits at which global symmetry breaking phenomena occur.
PubDate: 2017-12-26
DOI: 10.1007/s00526-017-1285-7
Issue No: Vol. 57, No. 1 (2017)

• Existence and phase separation of entire solutions to a pure critical
competitive elliptic system
• Authors: Mónica Clapp; Angela Pistoia
Abstract: Abstract We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic system \begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=\mu _{1} u ^{{2}^{*}-2}u+\lambda \alpha u ^{\alpha -2} v ^{\beta }u,\\ -\,\Delta v=\mu _{2} v ^{{2}^{*}-2}v+\lambda \beta u ^{\alpha } v ^{\beta {-2} }v,\\ u,v\in D^{1,2}({\mathbb {R}}^{N}), \end{array}\right. } \end{aligned} where $$N\ge 4,$$ $$2^{*}:=\frac{2N}{N-2}$$ is the critical Sobolev exponent, $$\alpha ,\beta \in (1,2],$$ $$\alpha +\beta =2^{*},$$ $$\mu _{1},\mu _{2}>0,$$ and $$\lambda <0.$$ We show that these solutions exhibit phase separation as $$\lambda \rightarrow -\,\infty ,$$ and we give a precise description of their limit domains. If $$\mu _{1}=\mu _{2}$$ and $$\alpha =\beta$$ , we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.
PubDate: 2017-12-26
DOI: 10.1007/s00526-017-1283-9
Issue No: Vol. 57, No. 1 (2017)

• Gradient flow for the Boltzmann entropy and Cheeger’s energy on
time-dependent metric measure spaces
• Authors: Eva Kopfer
Abstract: Abstract We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).
PubDate: 2017-12-23
DOI: 10.1007/s00526-017-1287-5
Issue No: Vol. 57, No. 1 (2017)

• Distances between classes in $$W^{1,1}(\Omega ;{\mathbb {S}}^{1})$$ W 1 ,
1 ( Ω ; S 1 )
• Authors: Haim Brezis; Petru Mironescu; Itai Shafrir
Abstract: Abstract We introduce an equivalence relation on the space $$W^{1,1}(\Omega ;{\mathbb {S}}^1)$$ which classifies maps according to their “topological singularities”. We establish sharp bounds for the distances (in the usual sense and in the Hausdorff sense) between the equivalence classes. Similar questions are examined for the space $$W^{1,p}(\Omega ;{\mathbb {S}}^1)$$ when $$p>1$$ .
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1280-z
Issue No: Vol. 57, No. 1 (2017)

• Flatness implies smoothness for solutions of the porous medium equation
• Authors: Clemens Kienzler; Herbert Koch; Juan Luis Vázquez
Abstract: Abstract One of the major problems in the theory of the porous medium equation $$\partial _t\rho =\Delta _x\rho ^m,\,m > 1$$ , is the regularity of the solutions $$\rho (t,x)\ge 0$$ and the free boundaries $$\Gamma =\partial \{(t,x): \rho >0\}$$ . Here we assume flatness of the solution and derive $$C^\infty$$ regularity of the interface after a small time, as well as $$C^\infty$$ regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time $$T_r$$ that depends on $$\rho _0$$ . More precisely, we prove that for $$t \ge T_r$$ the pressure $$\rho ^{m-1}$$ is $$C^\infty$$ in the positivity set and up to the free boundary, which is a $$C^\infty$$ hypersurface. Moreover, $$T_r$$ can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as $$t\rightarrow \infty$$ the solution increasingly resembles a Barenblatt function and the support looks like a ball.
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1296-4
Issue No: Vol. 57, No. 1 (2017)

• The Riesz transform and quantitative rectifiability for general Radon
measures
• Authors: Daniel Girela-Sarrión; Xavier Tolsa
Abstract: Abstract In this paper we show that if $$\mu$$ is a Borel measure in $${{\mathbb {R}}}^{n+1}$$ with growth of order n, such that the n-dimensional Riesz transform $${{\mathcal {R}}}_\mu$$ is bounded in $$L^2(\mu )$$ , and $$B\subset {{\mathbb {R}}}^{n+1}$$ is a ball with $$\mu (B)\approx r(B)^n$$ such that: there is some n-plane L passing through the center of B such that for some $$\delta >0$$ small enough, it holds \begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned} for some constant $${\varepsilon }>0$$ small enough, \begin{aligned}\int _{B} {{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1) ^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned} where $$m_{\mu ,B}({{\mathcal {R}}}_\mu 1)$$ stands for the mean of $${{\mathcal {R}}}_\mu 1$$ on B with respect to $$\mu$$ , then there exists a uniformly n-rectifiable set $$\Gamma$$ , with $$\mu (\Gamma \cap B)\gtrsim \mu (B)$$ , and such that $$\mu _\Gamma$$ is absolutely continuous with respect to $${{\mathcal {H}}}^n _\Gamma$$ . This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1294-6
Issue No: Vol. 57, No. 1 (2017)

• On the global well-posedness of 3-D axi-symmetric Navier–Stokes system
with small swirl component
• Authors: Yanlin Liu; Ping Zhang
Abstract: Abstract In this paper, we prove the local well-posedness of 3-D axi-symmetric Navier–Stokes system with initial data in the critical Lebesgue spaces. We also obtain the global well-posedness result with small initial data. Furthermore, with the initial swirl component of the velocity being sufficiently small in the almost critical spaces, we can still prove the global well-posedness of the system.
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1288-4
Issue No: Vol. 57, No. 1 (2017)

• A three-dimensional symmetry result for a phase transition equation in the
genuinely nonlocal regime
• Authors: Serena Dipierro; Alberto Farina; Enrico Valdinoci
Abstract: Abstract We consider bounded solutions of the nonlocal Allen–Cahn equation \begin{aligned} (-\Delta )^s u=u-u^3\qquad { \text{ in } }\mathbb {R}^3, \end{aligned} under the monotonicity condition  $$\partial _{x_3}u>0$$ and in the genuinely nonlocal regime in which  $$s\in \left( 0,\frac{1}{2}\right)$$ . Under the limit assumptions \begin{aligned} \lim _{x_n\rightarrow -\infty } u(x',x_n)=-1\quad { \text{ and } }\quad \lim _{x_n\rightarrow +\infty } u(x',x_n)=1, \end{aligned} it has been recently shown in Dipierro et al. (Improvement of flatness for nonlocal phase transitions, 2016) that u is necessarily 1D, i.e. it depends only on one Euclidean variable. The goal of this paper is to obtain a similar result without assuming such limit conditions. This type of results can be seen as nonlocal counterparts of the celebrated conjecture formulated by De Giorgi (Proceedings of the international meeting on recent methods in nonlinear analysis (Rome, 1978), Pitagora, Bologna, pp 131–188, 1979).
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1295-5
Issue No: Vol. 57, No. 1 (2017)

• $$T_5$$ T 5 -Configurations and non-rigid sets of matrices
• Authors: Clemens Förster; László Székelyhidi
Abstract: Abstract In 2003 B. Kirchheim-D. Preiss constructed a Lipschitz map in the plane with 5 incompatible gradients, where incompatibility refers to the condition that no two of the five matrices are rank-one connected. The construction is via the method of convex integration and relies on a detailed understanding of the rank-one geometry resulting from a specific set of five matrices. The full computation of the rank-one convex hull for this specific set was later carried out in 2010 by W. Pompe in Calc. Var. PDE 37(3–4):461–473, (2010) by delicate geometric arguments. For more general sets of matrices a full computation of the rank-one convex hull is clearly out of reach. Therefore, in this short note we revisit the construction and propose a new, in some sense generic method for deciding whether convex integration for a given set of matrices can be carried out, which does not require the full computation of the rank-one convex hull.
PubDate: 2017-12-22
DOI: 10.1007/s00526-017-1293-7
Issue No: Vol. 57, No. 1 (2017)

• Quasistatic evolution in non-associative plasticity revisited
• Authors: G. A. Francfort; M. G. Mora
Abstract: Abstract The mathematical treatment of evolutionary non-associative elasto-plasticity is still in its infancy. In particular, all existence results thus far rely on a spatially mollified stress admissibility constraint. Further, the evolution is formulated in a rescaled time from which it is very difficult to infer any useful information on the “real” time evolution. We propose a causal spatio-temporal mollification of the stress admissibility constraint that, while no more far-fetched than a purely spatial one, produces a more elegant and complete evolution for such models, and this in the “real” time variable.
PubDate: 2017-12-21
DOI: 10.1007/s00526-017-1284-8
Issue No: Vol. 57, No. 1 (2017)

• A concave–convex problem with a variable operator
• Authors: Alexis Molino; Julio D. Rossi
Abstract: Abstract We study the following elliptic problem $$-A(u) = \lambda u^q$$ with Dirichlet boundary conditions, where $$A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)$$ is the Laplacian in one part of the domain, $$D_1$$ , and the p-Laplacian (with $$p>2$$ ) in the rest of the domain, $$D_2$$ . We show that this problem exhibits a concave–convex nature for $$1<q<p-1$$ . In fact, we prove that there exists a positive value $$\lambda ^*$$ such that the problem has no positive solution for $$\lambda > \lambda ^*$$ and a minimal positive solution for $$0<\lambda < \lambda ^*$$ . If in addition we assume that p is subcritical, that is, $$p<2N/(N-2)$$ then there are at least two positive solutions for almost every $$0<\lambda < \lambda ^*$$ , the first one (that exists for all $$0<\lambda < \lambda ^*$$ ) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every $$0<\lambda < \lambda ^*$$ ) comes from an appropriate (and delicate) mountain pass argument.
PubDate: 2017-12-21
DOI: 10.1007/s00526-017-1291-9
Issue No: Vol. 57, No. 1 (2017)

• On Lipschitz continuity of solutions of hyperbolic Poisson’s
equation
• Authors: Jiaolong Chen; Manzi Huang; Antti Rasila; Xiantao Wang
Abstract: Abstract In this paper, we investigate solutions of the hyperbolic Poisson equation $$\Delta _{h}u(x)=\psi (x)$$ , where $$\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)$$ and \begin{aligned} \Delta _{h}u(x)= (1- x ^2)^2\Delta u(x)+2(n-2)\left( 1- x ^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned} is the hyperbolic Laplace operator in the n-dimensional space $$\mathbb {R}^n$$ for $$n\ge 2$$ . We show that if $$n\ge 3$$ and $$u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )$$ is a solution to the hyperbolic Poisson equation, then it has the representation $$u=P_{h}[\phi ]-G_{ h}[\psi ]$$ provided that $$u\mid _{\mathbb {S}^{n-1}}=\phi$$ and $$\int _{\mathbb {B}^{n}}(1- x ^{2})^{n-1} \psi (x) \,d\tau (x)<\infty$$ . Here $$P_{h}$$ and $$G_{h}$$ denote Poisson and Green integrals with respect to $$\Delta _{h}$$ , respectively. Furthermore, we prove that functions of the form $$u=P_{h}[\phi ]-G_{h}[\psi ]$$ are Lipschitz continuous.
PubDate: 2017-12-21
DOI: 10.1007/s00526-017-1290-x
Issue No: Vol. 57, No. 1 (2017)

• Regularity of harmonic maps from polyhedra to CAT(1) spaces
• Authors: Christine Breiner; Ailana Fraser; Lan-Hsuan Huang; Chikako Mese; Pam Sargent; Yingying Zhang
Abstract: Abstract We determine regularity results for energy minimizing maps from an n-dimensional Riemannian polyhedral complex X into a CAT(1) space. Provided that the metric on X is Lipschitz regular, we prove Hölder regularity with Hölder constant and exponent dependent on the total energy of the map and the metric on the domain. Moreover, at points away from the $$(n-2)$$ -skeleton, we improve the regularity to locally Lipschitz. Finally, for points $$x \in X^{(k)}$$ with $$k \le n-2$$ , we demonstrate that the Hölder exponent depends on geometric and combinatorial data of the link of $$x \in X$$ .
PubDate: 2017-12-21
DOI: 10.1007/s00526-017-1279-5
Issue No: Vol. 57, No. 1 (2017)

• On the Hölder continuous subsolution problem for the complex
Monge–Ampère equation
• Authors: Ngoc Cuong Nguyen
Abstract: Abstract We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge–Ampère equation admits Hölder continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge–Ampère total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902–930, 2016).
PubDate: 2017-12-20
DOI: 10.1007/s00526-017-1297-3
Issue No: Vol. 57, No. 1 (2017)

• Finite time blowup of the n -harmonic flow on n -manifolds
• Authors: Leslie Hon-Nam Cheung; Min-Chun Hong
Abstract: Abstract In this paper, we generalize the no-neck result of Qing and Tian (in Commun Pure Appl Math 50:295–310, 1997) to show that there is no neck during blowing up for the n-harmonic flow as $$t\rightarrow \infty$$ . As an application of the no-neck result, we settle a conjecture of Hungerbühler (in Ann Scuola Norm Sup Pisa Cl Sci 4:593–631, 1997) by constructing an example to show that the n-harmonic map flow on an n-dimensional Riemannian manifold blows up in finite time for $$n\ge 3$$ .
PubDate: 2017-12-20
DOI: 10.1007/s00526-017-1282-x
Issue No: Vol. 57, No. 1 (2017)

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