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Abstract: We suggest a new approach to solve a class of degenerate Hamilton–Jacobi equations without any assumptions on the emptiness of the Aubry set. It is based on the characterization of the maximal subsolution by means of the Fenchel–Rockafellar duality. This approach enables us to use augmented Lagrangian methods as alternatives to the commonly used methods for numerical approximation of the solution, based on finite difference approximation or on optimal control interpretation of the solution. PubDate: 2021-10-07

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Abstract: We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap V(x) rotating at the velocity \(\Omega \) . It is known that there exists a critical rotational velocity \(0<\Omega ^*:=\Omega ^*(V)\le \infty \) and a critical number \(0<a^*<\infty \) such that for any rotational velocity \(0\le \Omega <\Omega ^*\) , ground states exist if and only if the coupling constant a satisfies \(a<a^*\) . For a general class of traps V(x), which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as \(a\nearrow a^*\) , where \(\Omega \in (0,\Omega ^*)\) is fixed. PubDate: 2021-10-05

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Abstract: In this paper we study asymptotic behaviors of n-superharmonic functions at singularity using the Wolff potential and capacity estimates in nonlinear potential theory. Our results are inspired by and extend [6] of Arsove–Huber and [63] of Taliaferro in 2 dimensions. To study n-superharmonic functions we use a new notion of thinness in terms of n-capacity motivated by a type of Wiener criterion in [6]. To extend [63], we employ the Adams–Moser–Trudinger’s type inequality for the Wolff potential, which is inspired by the inequality used in [15] of Brezis–Merle. For geometric applications, we study the asymptotic end behaviors of complete conformally flat manifolds as well as complete properly embedded hypersurfaces in hyperbolic space. These geometric applications seem to elevate the importance of n-Laplace equations and make a closer tie to the classic analysis developed in conformal geometry in general dimensions. PubDate: 2021-09-22

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Abstract: We consider in this paper an area functional defined on submanifolds of fixed degree immersed into a graded manifold equipped with a Riemannian metric. Since the expression of this area depends on the degree, not all variations are admissible. It turns out that the associated variational vector fields must satisfy a system of partial differential equations of first order on the submanifold. Moreover, given a vector field solution of this system, we provide a sufficient condition that guarantees the possibility of deforming the original submanifold by variations preserving its degree. As in the case of singular curves in sub-Riemannian geometry, there are examples of isolated surfaces that cannot be deformed in any direction. When the deformability condition holds we compute the Euler–Lagrange equations. The resulting mean curvature operator can be of third order. PubDate: 2021-09-22

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Abstract: In this paper, we study the existence, uniqueness and properties of the Green’s function for the conic linear elliptic equation. As an application, we give a new proof of the Schauder estimate for conic Monge–Ampère equations. PubDate: 2021-09-22

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Abstract: We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an \(\mathsf{RCD}\) space. In this situation it appears that no version of the ‘subpartition lemma’ holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: the fact that such spaces are ‘strongly rectifiable’ a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim’s metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is \(\mathsf{CAT}(0)\) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation. PubDate: 2021-09-22

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Abstract: We derive a shape derivative formula for the family of principal Dirichlet eigenvalues \(\lambda _s(\Omega )\) of the fractional Laplacian \((-\Delta )^s\) associated with bounded open sets \(\Omega \subset \mathbb {R}^N\) of class \(C^{1,1}\) . This extends, with a help of a new approach, a result in Dalibard and Gérard-Varet (Calc. Var. 19(4):976–1013, 2013) which was restricted to the case \(s=\frac{1}{2}\) . As an application, we consider the maximization problem for \(\lambda _s(\Omega )\) among annular-shaped domains of fixed volume of the type \(B\setminus \overline{B}'\) , where B is a fixed ball and \(B'\) is ball whose position is varied within B. We prove that \(\lambda _s(B\setminus \overline{B}')\) is maximal when the two balls are concentric. Our approach also allows to derive similar results for the fractional torsional rigidity. More generally, we will characterize one-sided shape derivatives for best constants of a family of subcritical fractional Sobolev embeddings. PubDate: 2021-09-22

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Abstract: We study the Ginzburg–Landau functional describing an extreme type-II superconductor wire with cross section with finitely many corners at the boundary. We derive the ground state energy asymptotics up to o(1) errors in the surface superconductivity regime, i.e., between the second and third critical fields. We show that, compared to the case of smooth domains, each corner provides an additional contribution of order \( {\mathcal {O}}(1) \) depending on the corner opening angle. The corner energy is in turn obtained from an implicit model problem in an infinite wedge-like domain with fixed magnetic field. We also prove that such an auxiliary problem is well-posed and its ground state energy bounded and, finally, state a conjecture about its explicit dependence on the opening angle of the sector. PubDate: 2021-09-22

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Abstract: We perform a fast-reaction limit for a linear reaction-diffusion system consisting of two diffusion equations coupled by a linear reaction. We understand the linear reaction-diffusion system as a gradient flow of the free energy in the space of probability measures equipped with a geometric structure, which contains the Wasserstein metric for the diffusion part and cosh-type functions for the reaction part. The fast-reaction limit is done on the level of the gradient structure by proving EDP-convergence with tilting. The limit gradient system induces a diffusion system with Lagrange multipliers on the linear slow-manifold. Moreover, the limit gradient system can be equivalently described by a coarse-grained gradient system, which induces a diffusion equation with a mixed diffusion constant for the coarse-grained slow variable. PubDate: 2021-09-19

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Abstract: We study the formation of singularities for cylindrical symmetric solutions to the Gross–Pitaevskii equation describing a dipolar Bose–Einstein condensate. We prove that solutions arising from initial data with energy below the energy of the Ground State and that do not scatter collapse in finite time. The main tools to prove our result are the variational characterization of the Ground State energy, suitable localized virial identities for cylindrical symmetric functions, and general integral and pointwise estimates for operators involving powers of the Riesz transform. PubDate: 2021-09-19

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Abstract: We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau–de Gennes Q-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems. PubDate: 2021-09-19

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Abstract: Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points. PubDate: 2021-09-19

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Abstract: In this paper, we derive a priori interior Hessian estimates for the Lagrangian mean curvature equation if the Lagrangian phase is supercritical and has bounded second derivatives. PubDate: 2021-09-19

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Abstract: We consider the Dirichlet problem of the Stokes equations in a domain with a shrinking hole in \({\mathbb {R}}^d, \ d\ge 2\) . A typical observation is that, the Lipschitz norm of the domain goes to infinity as the size of the hole goes to zero. Thus, if \(p\ne 2\) , the classical results indicate that the \(W^{1,p}\) estimate of the solution may go to infinity as the size of the hole tends to zero. With the presence of the shrinking hole in a fixed domain, we give a complete description for the uniform \(W^{1,p}\) estimates of the solution for all \(1<p<\infty \) . We show that the uniform \(W^{1,p}\) estimate holds if and only if \(d'<p<d\) ( \(p=2\) when \(d=2\) ). We then give two applications in the study of homogenization problems in fluid mechanics: a generalization of the restriction operator and a construction of Bogovskii type operator in perforated domains with a quantitative estimate of the operator norm. PubDate: 2021-09-19

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Abstract: We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We provide a counterexample to the conjecture. We generalise the Kantorovich–Rubinstein duality to the vector measures setting. Employing the generalisation, we answer the conjecture in the affirmative provided there exists an optimal transport with absolutely continuous marginals of its total variation. PubDate: 2021-09-19

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Abstract: Given a constant \(k>1\) , let Z be the family of round spheres of radius \({{\,\mathrm{artanh}\,}}(k^{-1})\) in the hyperbolic space \({\mathbb {H}}^3\) , so that any sphere in Z has mean curvature k. We prove a crucial nondegeneracy result involving the manifold Z. As an application, we provide sufficient conditions on a prescribed function \(\phi \) on \({\mathbb {H}}^3\) , which ensure the existence of a \(\mathcal{C}^1\) -curve, parametrized by \(\varepsilon \approx 0\) , of embedded spheres in \({\mathbb {H}}^3\) having mean curvature \(k +\varepsilon \phi \) at each point. PubDate: 2021-09-16

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Abstract: We revisit the liquid drop model with a general Riesz potential. Our new result is the existence of minimizers for the conjectured optimal range of parameters. We also prove a conditional uniqueness of minimizers and a nonexistence result for heavy nuclei. PubDate: 2021-09-16

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Abstract: We provide perturbative estimates for the one-phase Stefan free boundary problem and obtain the regularity of flat free boundaries via a linearization technique in the spirit of the elliptic counterpart established in De Silva (IFB 13, 223–238, 2011). PubDate: 2021-09-04

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Abstract: We present a proof of strict g-convexity in 2D for solutions of generated Jacobian equations with a g-Monge–Ampère measure bounded away from 0. Subsequently this implies \(C^1\) differentiability in the case of a g-Monge–Ampère measure bounded from above. Our proof follows one given by Trudinger and Wang in the Monge–Ampère case. Thus, like theirs, our argument is local and yields a quantitative estimate on the g-convexity. As a result our differentiability result is new even in the optimal transport case: we weaken previously required domain convexity conditions. Moreover in the optimal transport case and the Monge–Ampère case our key assumptions, namely A3w and domain convexity, are necessary. PubDate: 2021-09-04

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Abstract: We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve. PubDate: 2021-09-04