Conormal Boundary Research Articles

A Note on Partial Quasi-Morphisms and Products in Lagrangian Floer Homology in Cotangent Bundles

We define partial quasi-morphisms on the group of Hamiltonian diffeomorphisms of the cotangent bundle using the spectral invariants in Lagrangian Floer homology with conormal boundary conditions, where the product compatible with the PSS isomorphism and the homological intersection product is lacking.

On conormal derivative problem for parabolic equations with Dini mean oscillation coefficients

<p style='text-indent:20px;'>We show that weak solutions to parabolic equations in divergence form with conormal boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain integrability conditions.

A New Approach for Euler-Lagrange Orbits on Compact Manifolds with Boundary

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two.

Time Analyticity for Inhomogeneous Parabolic Equations and the Navier–Stokes Equations in the Half Space

We prove the time analyticity for weak solutions of inhomogeneous parabolic equations with measurable coefficients in the half space with either the Dirichlet boundary condition or the conormal boundary condition under the assumption that the solution and the source term have the exponential growth of order $2$ with respect to the space variables. We also obtain the time analyticity for bounded mild solutions of the incompressible Navier-Stokes equations in the half space with the Dirichlet boundary condition. Our work is an extension of the recent work by Zhang [Proc. Amer. Math. Soc. 148 (2020)] and Dong-Zhang [J. Funct. Anal. (2020)], where the authors proved the time analyticity of solutions to the homogeneous heat equation and the Navier-Stokes equations in the whole space.

A multiplicity result for Euler–Lagrange orbits satisfying the conormal boundary conditions

In this paper, we study the multiplicity problem for Euler–Lagrange orbits that satisfy the conormal boundary conditions for a suitable class of reversible Lagrangian functions on compact manifolds. Such a class contains, e.g. the energy function of reversible Finsler metrics that satisfy a convexity condition on the boundary.

Higher integrability result for nonlinear elliptic systems with conormal boundary conditions

We prove a boundary higher integrability result for nonlinear elliptic systems with conormal boundary conditions in locally uniform domains. To do it, we derive a boundary version of Gehring–Giaquinta–Modica Lemma and Sobolev–Poincaré type inequality in locally uniform domains. Our result plays a key role for handling the coefficients when obtaining Calderó n–Zygmund type estimates with conormal boundary conditions in nonsmooth domains.

Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for $$p >1$$ . Under some sufficient conditions, we establish local interior, local boundary, and global $$W^{1,q}$$ -regularity estimates for weak solutions with $$q>p$$ , assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.

On continuity equations in space-time domains

In this paper we consider a class of continuity equations that are conditioned to stay in general space-time domains, which is formulated as a continuum limit of interacting particle systems. Firstly, we study the well-posedness of the solutions and provide examples illustrating that the stability of solutions is strongly related to the decay of initial data at infinity. In the second part, we consider the vanishing viscosity approximation of the system, given with the co-normal boundary data. If the domain is spatially convex, the limit coincides with the solution of our original system, giving another interpretation to the equation.

On the existence of Euler–Lagrange orbits satisfying the conormal boundary conditions

Let (M,g) be a closed connected Riemannian manifold, L:TM→R be a Tonelli Lagrangian. Given two non-empty closed submanifolds Q0,Q1⊆M and a real number k, we study the existence of Euler–Lagrange orbits with energy k connecting Q0 to Q1 and satisfying suitable boundary conditions, known as conormal boundary conditions. We introduce the Mañé critical value which is relevant for this problem and discuss existence results for supercritical and subcritical energies. We also provide counterexamples showing that all the results are sharp.

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary. Our result is the nonlinear version of the classical result in [3] for divergence-form operators with co-normal boundary data. The main ingredients in our analysis are the estimate on the oscillation on the solutions in half-spaces (Theorem 3.1), and the estimate on the mode of convergence of the solutions as the normal of the half-space varies over irrational directions (Theorem 4.1).

On Multiplicity of Positive Solutions for Quasilinear Equation with Co-normal Boundary Condition

Abstract Let Ω ⊂ ℝN, N ≥ 3, be a bounded domain with C2 boundary, , the critical exponent for the Sobolev imbedding. In this work, we are interested in the following problem: where λ &gt; 0, 0 ≤ q &lt; p - 1. We show that there exists 0 &lt; Λ &lt; ∞ such that for suitable ranges of p and q, (Pλ) admits at least two solutions in W1,p (Ω) if λ ∈ (0, Λ) and no solution if λ &gt; Λ. The proof of these assertions is done by first finding the local minimum for the variational functional associated to (Pλ) and then applying mountain pass arguments to obtain a saddle point type solution. In the critical case we are considering, there are technical reasons which make the mountain pass argument work for only certain ranges of p and q.

The M-matrix inverse problem for the Sturm—Liouville equation on graphs

We consider an inverse spectral problem for Sturm–Liouville boundary-value problems on a graph with formally self-adjoint boundary conditions at the nodes, where the given information is the M-matrix. Based on the authors' previous results, using Green's function, we prove that the poles of the M-matrix are at the eigenvalues of the associated boundary-value problem and are simple, located on the real axis, and that the residue at a pole is a negative semi-definite matrix with rank equal to the multiplicity of the eigenvalue. We define the so-called norming constants and relate them to the spectral measure and the M-matrix. This enables us to recover, from the M-matrix, the boundary conditions and the potential, up to a unitary equivalence for co-normal boundary conditions.

The homology of path spaces and Floer homology with conormal boundary conditions

We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this chain complex is isomorphic to the singular homology of the natural path space associated to the boundary conditions.

BV functions and parabolic initial boundary value problems on domains

Given a uniformly elliptic second order operator \({\mathcal{A}}\) on a possibly unbounded domain \({\Omega\,\subset\,\mathbb {R}^N}\) , let (T(t))t≥0 be the semigroup generated by \({\mathcal{A}}\) in L1(Ω), under homogeneous co-normal boundary conditions on ∂Ω. We show that the limit as t → 0 of the L1-norm of the spatial gradient DxT(t)u0 tends to the total variation of the initial datum u0, and in particular is finite if and only if u0 belongs to BV(Ω). This result is true also for weighted BV spaces. A further characterization of BV functions in terms of the short-time behaviour of (T(t))t≥0 is also given.

Time-periodic solutions of quasilinear parabolic differential equations III: conormal boundary conditions

Time-periodic solutions of quasilinear parabolic differential equations III: conormal boundary conditions

Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions

We study solutions of the conormal derivative problem for uniformly parabolic equations in divergence form. Under weak regularity hypotheses on the operator, the global Holder continuity of the gradient of a weak solution is established. The method of proof is based on [5] and the results extend those in [7, Section V.7].

Correction to «Holder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions»

In the proof of [1, Theorem 1.2] an incorrect assertion was made. Using the notation of that work, we claimed that [D(u)−v]δis finite, which may not be true. We give here a correct, simpler proof of the theorem.

Capillary surfaces over obstacles

We consider the usual capillarity problem with the additional requirement that the capillary surface lies above some obstacle. This involves a variational inequality instead of a boundary value problem. We prove existence of a solution to the variational inequality and study the boundary regularity. In particular, global C 11 -regularity is shown for a wider class of variational inequalities with conormal boundary condition.