513 research outputs found
A threshold phenomenon for embeddings of into Orlicz spaces
We consider a sequence of positive smooth critical points of the
Adams-Moser-Trudinger embedding of into Orlicz spaces. We study its
concentration-compactness behavior and show that if the sequence is not
precompact, then the liminf of the -norms of the functions is greater
than or equal to a positive geometric constant.Comment: 14 Page
Small coupling limit and multiple solutions to the Dirichlet Problem for Yang Mills connections in 4 dimensions - Part I
In this paper (Part I) and its sequels (Part II and Part III), we analyze the
structure of the space of solutions to the epsilon-Dirichlet problem for the
Yang-Mills equations on the 4-dimensional disk, for small values of the
coupling constant epsilon. These are in one-to-one correspondence with
solutions to the Dirichlet problem for the Yang Mills equations, for small
boundary data. We prove the existence of multiple solutions, and, in
particular, non minimal ones, and establish a Morse Theory for this non-compact
variational problem. In part I, we describe the problem, state the main
theorems and do the first part of the proof. This consists in transforming the
problem into a finite dimensional problem, by seeking solutions that are
approximated by the connected sum of a minimal solution with an instanton, plus
a correction term due to the boundary. An auxiliary equation is introduced that
allows us to solve the problem orthogonally to the tangent space to the space
of approximate solutions. In Part II, the finite dimensional problem is solved
via the Ljusternik-Schirelman theory, and the existence proofs are completed.
In Part III, we prove that the space of gauge equivalence classes of Sobolev
connections with prescribed boundary value is a smooth manifold, as well as
some technical lemmas used in Part I. The methods employed still work when the
4-dimensional disk is replaced by a more general compact manifold with
boundary, and SU(2) is replaced by any compact Lie group
Constant sign and nodal solutions for a class of nonlinear Dirichlet problems
We consider a nonlinear Dirichlet problem with a Carathéodory reaction which has arbitrary growth from below. We show that the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal. In the semilinear case (i.e., p =2), with the reaction f(z, .)being C1and with subcritical growth, we show that there is a second nodal solution, for a total of four nontrivial smooth solutions. Finally,when the reaction has concave terms and is subcritical and for the nonlinear problem (i.e., 1 <p <∞) we show that again we can have the existence of three nontrivial smooth solutions, two of constant sign and a third nodal
Asymptotic stability, concentration, and oscillation in harmonic map heat-flow, Landau-Lifshitz, and Schroedinger maps on R^2
We consider the Landau-Lifshitz equations of ferromagnetism (including the
harmonic map heat-flow and Schroedinger flow as special cases) for degree m
equivariant maps from R^2 to S^2. If m \geq 3, we prove that near-minimal
energy solutions converge to a harmonic map as t goes to infinity (asymptotic
stability), extending previous work down to degree m = 3. Due to slow spatial
decay of the harmonic map components, a new approach is needed for m=3,
involving (among other tools) a "normal form" for the parameter dynamics, and
the 2D radial double-endpoint Strichartz estimate for Schroedinger operators
with sufficiently repulsive potentials (which may be of some independent
interest). When m=2 this asymptotic stability may fail: in the case of
heat-flow with a further symmetry restriction, we show that more exotic
asymptotics are possible, including infinite-time concentration (blow-up), and
even "eternal oscillation".Comment: 34 page
A general wavelet-based profile decomposition in the critical embedding of function spaces
We characterize the lack of compactness in the critical embedding of
functions spaces having similar scaling properties in the
following terms : a sequence bounded in has a subsequence
that can be expressed as a finite sum of translations and dilations of
functions such that the remainder converges to zero in as
the number of functions in the sum and tend to . Such a
decomposition was established by G\'erard for the embedding of the homogeneous
Sobolev space into the in dimensions with
, and then generalized by Jaffard to the case where is a Riesz
potential space, using wavelet expansions. In this paper, we revisit the
wavelet-based profile decomposition, in order to treat a larger range of
examples of critical embedding in a hopefully simplified way. In particular we
identify two generic properties on the spaces and that are of key use
in building the profile decomposition. These properties may then easily be
checked for typical choices of and satisfying critical embedding
properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older
and BMO spaces.Comment: 24 page
\epsilon-regularity for systems involving non-local, antisymmetric operators
We prove an epsilon-regularity theorem for critical and super-critical
systems with a non-local antisymmetric operator on the right-hand side.
These systems contain as special cases, Euler-Lagrange equations of
conformally invariant variational functionals as Rivi\`ere treated them, and
also Euler-Lagrange equations of fractional harmonic maps introduced by Da
Lio-Rivi\`ere.
In particular, the arguments presented here give new and uniform proofs of
the regularity results by Rivi\`ere, Rivi\`ere-Struwe, Da-Lio-Rivi\`ere, and
also the integrability results by Sharp-Topping and Sharp, not discriminating
between the classical local, and the non-local situations
On completeness of orbits of Killing vector fields
A Theorem is proved which reduces the problem of completeness of orbits of
Killing vector fields in maximal globally hyperbolic, say vacuum, space--times
to some properties of the orbits near the Cauchy surface. In particular it is
shown that all Killing orbits are complete in maximal developements of
asymptotically flat Cauchy data, or of Cauchy data prescribed on a compact
manifold. This result gives a significant strengthening of the uniqueness
theorems for black holes.Comment: 16 pages, Latex, preprint NSF-ITP-93-4
Green's functions for parabolic systems of second order in time-varying domains
We construct Green's functions for divergence form, second order parabolic
systems in non-smooth time-varying domains whose boundaries are locally
represented as graph of functions that are Lipschitz continuous in the spatial
variables and 1/2-H\"older continuous in the time variable, under the
assumption that weak solutions of the system satisfy an interior H\"older
continuity estimate. We also derive global pointwise estimates for Green's
function in such time-varying domains under the assumption that weak solutions
of the system vanishing on a portion of the boundary satisfy a certain local
boundedness estimate and a local H\"older continuity estimate. In particular,
our results apply to complex perturbations of a single real equation.Comment: 25 pages, 0 figur
Multiple solutions to a magnetic nonlinear Choquard equation
We consider the stationary nonlinear magnetic Choquard equation
[(-\mathrm{i}\nabla+A(x))^{2}u+V(x)u=(\frac{1}{|x|^{\alpha}}\ast |u|^{p})
|u|^{p-2}u,\quad x\in\mathbb{R}^{N}%] where is a real valued vector
potential, is a real valued scalar potential ,
and . \ We assume that both and are
compatible with the action of some group of linear isometries of
. We establish the existence of multiple complex valued
solutions to this equation which satisfy the symmetry condition where
is a given group homomorphism into the unit
complex numbers.Comment: To appear on ZAM
Singular kernels, multiscale decomposition of microstructure, and dislocation models
We consider a model for dislocations in crystals introduced by Koslowski,
Cuiti\~no and Ortiz, which includes elastic interactions via a singular kernel
behaving as the norm of the slip. We obtain a sharp-interface limit
of the model within the framework of -convergence. From an analytical
point of view, our functional is a vector-valued generalization of the one
studied by Alberti, Bouchitt\'e and Seppecher to which their rearrangement
argument no longer applies. Instead we show that the microstructure must be
approximately one-dimensional on most length scales and exploit this property
to derive a sharp lower bound
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