This article is concerned with the problem of prescribing Gaussian curvature K and geodesic curvature h in a compact surface with boundary with conical singularities {p1, ... , pn} and corners {q1, ... , qn}. This is equivalent to solving the Liouville-type equation: { - Delta u+2k(0) = 2ke(u) - 4 pi Sigma (n)( i=1)alpha( i) (delta( pi) - 1 /|Sigma|) -2 pi Sigma (m )(j=1) beta(j) (delta(qi) -1 /|Sigma|) in Sigma partial derivative(v)u+2h(0) = 2he (u /2) on partial derivative Sigma where K0, h0 are the pre-existing Gaussian curvature and the geodesic curvature, respectively, and alpha(i), beta(j )> -1 are given. Solutions are obtained using a new variational formulation, first introduced in Thierry (1998) in [1] for the regular counterpart of the problem and extended here to the singular case. As far as we know, this is the first result for the problem of prescribed curvatures in surfaces with the two types of singularities. Key ingredients are a blow-up analysis around a sequence of points different from local maxima and Morse index estimates.
This article is concerned with the interaction between a planar traveling front and a compact obstacle for monotone bistable reaction-diffusion systems in exterior domains. By constructing appropriate sub- and supersolutions, we first establish the existence, uniqueness and monotonicity of the entire solution emanating from a planar traveling front. In particular, we verify that regardless of the shape of the obstacle, the entire solution locally converges to a stationary solution as time tends to infinity. Under the complete propagation assumption, we further show that the entire solution recovers to the same planar traveling front as time tends to infinity after passing the obstacle, and it constitutes a transition front. In addition, we provide some geometric conditions on the obstacle to ensure that the complete propagation assumption is nonempty. Without the complete propagation assumption, we prove that the entire solution still propagates in the form of the planar traveling front far behind the obstacle for large time. Finally, we apply our theoretical results to the Lotka-Volterra competition-diffusion system and the buffered bistable system.
It is well known, in the acoustic model, that highly contrasting transmission leads to the so-called Minnaert subwavelength resonance. In this work, we show that such highly contrasting transmissions create not only one resonance but a family of infinite resonances located near the real axis where the first one (i.e. the smallest) is indeed the Minnaert one. This family of resonances are the shifts (in the lower complex plan) of the Neumann eigenvalues of the Laplacian. The well known Minneart resonance is nothing but the shift of the trivial (zero) Neumann eigenvalue of the bubble. These resonances, other than the Minnaert ones, are Fabry-Pérot-type resonances as the generated total fields, in the bubble, are dominated by a linear combination of the Neumann eigenfunctions which, in particular, might create interferences. In addition, we establish the following properties. 1. We derive the asymptotic expansions, at the second order, of this family of resonances in terms of the contrasting coefficient. 2. In the time-harmonic regime, we derive the resolvent estimates of the related Hamiltonian and the asymptotics of scattered fields that are uniform in the whole space, highlighting the contributions from this sequence of resonances. 3. In the time domain regime, we derive the time behavior of the acoustic microresonator at large time-scales inversely proportional to powers of microresonator's radius. 4. The analysis shows that near Fabry-Pérot resonances, the mircoresonator exhibits pronounced anisotropy. We believe that such a feature may pave the way for designing anisotropic metamaterials from simple configurations of a single microresonator.
In this paper, we investigate the borderline regularity of local minimizers of energy functionals under minimal assumptions on the potential term σ. When σ is merely bounded and measurable, we show that sign-changing minimizers are Log-Lipschitz continuous, which represents the optimal regularity in this general setting. In the one-phase case, however, we establish gradient bounds for minimizers along their free boundaries, revealing a structural gain in regularity. Most notably, we prove that if σ is continuous, then minimizers are of class C^1 along the free boundary, thereby identifying a sharp threshold for differentiability in terms of the regularity of the potential.
In this manuscript, we provide local L^q-estimates for the gradient of solutions of a class of quasilinear equations whose principal part lacks strong monotonicity. These estimates are used to establish uniform large-scale L^q-estimates for the gradient of solutions of degenerate/singular quasilinear equations with oscillating coefficients and large-scale Lipschitz estimates for solutions of non-degenerate equations.
Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-H & eacute;non equation involving a quasi-linear degenerate diffusion and a space-dependent superlinear source featuring a singular potential partial derivative(t)u = Delta u(m) + |x|(sigma)u(p), t > 0, x is an element of R-N, when m > 1, p > 1 and sigma is an element of (max{-2, -N}, 0). While the superlinear source induces finite time blow-up when sigma = 0, whatever the value of p > 1, at least for sufficiently large initial conditions, a striking effect of the singular potential |x|(sigma) is the prevention of finite time blow-up for suitably small values of p, namely, 1 < p <= p(G) := [2-sigma(m-1)]/2. Such a result, as well as the local existence of solutions for p > p(G), is obtained by employing the Caffarelli-Kohn-Nirenberg inequalities. Another interesting feature is that uniqueness and comparison principle hold true for generic non-negative initial conditions when p > pG, but their validity is restricted to initial conditions which are positive in a neighborhood of x = 0 when p is an element of (1, p(G)), a range in which non-uniqueness holds true without this positivity condition. Finite time blow-up of any non-trivial, non-negative solution is established when pG < p <= p(F) := m + (sigma + 2)/N, while global existence for small initial data in some critical Lebesgue spaces and blow-up in finite time for initial data with a negative energy are proved for p > p(F).Optimal temporal growth rates are also derived for global solutions when p is an element of (1, p(G)]. All the results are sharp with respect to the exponents (m, p, sigma) and conditions on u(0).
Recently, it was demonstrated that, if it exists, the limit free energy of possibly non-convex spin glass models must be determined by a characteristic of the associated infinite-dimensional non-convex Hamilton-Jacobi equation. In this work, we investigate a similar theme purely from the perspective of PDEs. Specifically, we study the unique viscosity solution of the aforementioned equation and derive an envelope-type representation formula for the solution, in the form proposed by Evans. The value of the solution is expressed as an average of the values along characteristic lines, weighted by a non-explicit probability measure. The technical challenges arise not only from the infinite dimensionality but also from the fact that the equation is defined on a closed convex cone with an empty interior, rather than on the entire space. In the introduction, we provide a description of the motivation from spin glass theory and present the corresponding results for comparison with the PDE results.
We consider the existence and Lq gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in & Ropf;n, n >= 3. The first two results assume the drift is either in Ln or sufficiently small in weak Ln. The third result assumes the drift is in weak Ln without smallness, and obtain results for q close to 2.
We consider the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 [math.AP]. The equation models aggregation-diffusion phenomena with advection that is discontinuous and depends sharply on the gradient of the density itself. The quasi-linearity of the problem poses major challenges in the construction of the solution and complications arise in the proof of regularity. Our method overcomes these obstacle by relying solely on entropy inequalities and the theory of monotone operators. We provide existence, uniqueness and smoothing estimates in any dimensional space.
We derive logarithmic gradient estimate and universal boundedness estimate for semilinear elliptic equations on RCD*(K,N) metric measure spaces, which contains the class of Riemannian manifolds with Ricci curvature bounded below. These estimates are applicable for equations satisfying subcritical index condition, which recover many classical results even on Euclidean spaces. In certain case, these estimates are optimal even on RCD*(K,N) spaces with K<0. Two direct corollaries of these estimates are Harnack inequality and Liouville theorem. In addition to these estimates, we also establish fundamental relations among the universal boundedness estimate, the logarithmic gradient estimate, and Harnack inequality. Under certain and wild assumptions for the nonlinear term, we prove that these estimates are kappa-equivalent on RCD*(0,N) spaces for any kappa>1.
We study the asymptotic stability of the sine-Gordon kinks under small perturbations in weighted Sobolev norms. Our main tool is the B & auml;cklund transform which reduces the study of the asymptotic stability of the kinks to the study of the asymptotic decay of solutions near zero. Our results consist of two parts. First, we prove an asymptotic stability result similar to the local results in Alejo et al. and Chen et al. Our assumptions are the same as those in the local result in Chen et al. In its proof, we apply a result obtained by the inverse scattering method on the local decay of the solutions with sufficiently small and localized initial data. Moreover, we derive an asymptotic formula for the perturbations, i.e., the difference between solutions and kinks. This result is similar to that in L & uuml;hrmann and Schlag and the full asymptotic stability result in Chen et al. In its proof, we apply a result obtained by the method of testing by wave packets on the pointwise decay of the solutions with small and localized data.
We study the minimizing movement scheme for families of geodesically semi-convex functionals defined on either the Hellinger-Kantorovich or the Spherical Hellinger-Kantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves, which are produced by geodesically interpolating the points generated by the minimizing movement scheme, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the time step goes to 0.
We consider a class of hyperbolic systems that can be interpreted as approximations of the relativistic Vlasov-Maxwell system. These equations are derived by taking into account radiation-reaction effects occurring at a microscopic level. They involve a small length & ell;is an element of R+& lowast;, called the Bopp-Podolsky parameter. In this context, we address common issues in kinetic equations, such as the propagation of moments, regularity properties, and well-posedness. We also investigate semiclassical limits appearing when & ell; goes to zero.
While there are various results on the long-time behavior of the Willmore flow, the Helfrich flow with non-zero spontaneous curvature as its natural generalization is not yet well-understood. Past results for the gradient flow of a locally area- and volume-constrained Willmore flow indicate the existence of finite-time singularities which correspond to the scaling-behavior of the underlying energy. However, for a non-vanishing spontaneous curvature, the scaling behavior is not quite as conclusive. Indeed, in this article, we find that a negative spontaneous curvature corresponds to finite-time singularities of the locally constrained Helfrich flow if the initial surface is close to a round sphere in terms of its Willmore energy. Conversely however, in the case of a positive spontaneous curvature, we find a positive result in terms of the convergence behavior: The locally area-constrained Helfrich flow starting from a spherical immersion with suitably small Helfrich energy exists globally and converges to a Helfrich immersion after reparametrization. Moreover, this energetic smallness assumption is given by an explicit energy threshold depending on the spontaneous curvature and the local area constraint of the energy.
We investigate the Cauchy problem of three-dimensional compressible viscous and heat-conducting micropolar fluid equations with far-field vacuum. Based on delicate energy estimates and the structure of the system under consideration, we show global strong solutions for such a model under some smallness condition. In particular, our smallness condition is independent of any initial data except the initial mass.
We revisit a Harnack inequality for antisymmetric functions that has been recently established for the fractional Laplacian and we extend it to more general nonlocal elliptic operators. The new approach to deal with these problems that we propose in this paper leverages Bochner's relation, allowing one to relate a one-dimensional Fourier transform of an odd function with a three-dimensional Fourier transform of a radial function. In this way, Harnack inequalities for odd functions, which are essentially Harnack inequalities of boundary type, are reduced to interior Harnack inequalities.
We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow.
Well-posedness and higher regularity of the heat equation with Robin boundary conditions in an unbounded two-dimensional wedge is established in an L^2-setting of monomially weighted spaces. A mathematical framework is developed which allows to obtain arbitrarily high regularity without a smallness assumption on the opening angle of the wedge. The challenging aspect is that the resolvent problem exhibits two breakings of the scaling invariance, one in the equation and one in the boundary condition.
We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations, (Equation1.1), with initial velocity u(0) being sufficiently small in <(B)over dot>(1/2)(2,infinity) and with initial density being bounded from above and below. We first prove the global existence of the Fujita-Kato solution to the system (Equation1.1) if we assume in addition that the initial velocity u(0) is an element of<(H)over dot>(1/2). While under the additional assumptions that the initial velocity u(0) is an element of<(B)over dot>(1/2)(2,1) and initial density rho(0) satisfying rho(-1)(0)-1 is an element of<(B)over dot>(6,1), we prove that parallel to rho-1-1 parallel to((L) over tilde infinity(R+;<(B)over dot>1/26,1)) and parallel to u parallel to((L) over tilde infinity(R+;<(B)over dot>1/22,1)boolean AND L1(R+<(B)over dot>5/22,1)) are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system (Equation1.1) but also improve the exponential-in-time growth estimate for the solution in [Citation1] to be the uniform-in-time estimate (Equation1.11).
