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We consider the energy-critical harmonic heat flow from into a smooth compact revolution surface of . For initial data with corotational symmetry, the evolution reduces to the semilinear radially symmetric parabolic problem
for a suitable class of functions . Given an integer , we exhibit a set of initial data arbitrarily close to the least energy harmonic map in the energy-critical topology such that the corresponding solution blows up in finite time by concentrating its energy
at a speed given by the quantized rates
in accordance with the formal predictions of van den Berg et al. (2003). The case corresponds to the stable regime exhibited in our previous work (CPAM, 2013), and the data for leave on a manifold of codimension in some weak sense. Our analysis is a continuation of work by Merle, Rodnianski, and the authors (in various combinations) and it further exhibits the mechanism for the existence of the excited slow blow-up rates and the associated instability of these threshold dynamics.
where is the unitary ball and is Sobolev-subcritical. Such a problem arises in the search for solitary wave solutions for nonlinear Schrödinger equations (NLS) with power nonlinearity on bounded domains. Necessary and sufficient conditions (about , and ) are provided for the existence of solutions. Moreover, we show that standing waves associated to least energy solutions are orbitally stable for every (in the existence range) when is -critical and subcritical, i.e., , while they are stable for almost every in the -supercritical regime . The proofs are obtained in connection with the study of a variational problem with two constraints of independent interest: to maximize the -norm among functions having prescribed - and -norms.
We prove that for mappings in , continuous up to the boundary and with modulus of continuity satisfying a certain divergence condition, the image of the boundary of the unit ball has zero -Hausdorff measure. For Hölder continuous mappings we also prove an essentially sharp generalised Hausdorff dimension estimate.
We consider the problem of extending functions to functions for . We assume belongs to the critical space and we construct a -controlled extension . The Lorentz–Sobolev space is optimal for such controlled extension. Then we use these results to construct global controlled gauges for -connections over trivial -bundles in dimensions. This result is a global version of the local Sobolev control of connections obtained by K. Uhlenbeck.
Given a three-dimensional Riemannian manifold , we prove that, if is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres) having Willmore energy bounded above uniformly strictly by and Hausdorff converging to a point , then and (respectively, ). Moreover, a suitably rescaled sequence smoothly converges, up to subsequences and reparametrizations, to a round sphere in the euclidean three-dimensional space. This generalizes previous results of Lamm and Metzger. An application to the Hawking mass is also established.
In this paper, we study hole probabilities of Gaussian random polynomials of degree over a polydisc . When , we find asymptotic formulas and the decay rate of . In dimension one, we also consider hole probabilities over some general open sets and compute asymptotic formulas for the generalized hole probabilities over a disc .
We present stochastic homogenization results for viscous Hamilton–Jacobi equations using a new argument that is based only on the subadditive structure of maximal subsolutions (i.e., solutions of the “metric problem”). This permits us to give qualitative homogenization results under very general hypotheses: in particular, we treat nonuniformly coercive Hamiltonians that satisfy instead a weaker averaging condition. As an application, we derive a general quenched large deviation principle for diffusions in random environments and with absorbing random potentials.
We prove global existence of smooth solutions for a slightly supercritical hyperdissipative Navier–Stokes under the optimal condition on the correction to the dissipation. This proves a conjecture formulated by Tao.