Mathematics

Let u be a harmonic function in a C 1 -Dini domain D such that u vanishes on a boundary surface ball ?�D??B 5R (0) . We consider an effective version of its singular set (up to boundary) S(u):={X??D ¯ ¯ ¯ ¯ :u(X)=|?�u(X)|=0} and give an estimate of its (d??) -dimensional Minkowski content, which only depends on the upper bound of some modified frequency function of u centered at 0 . Such results are already known in the interior and at the boundary of convex domains, when the standard frequency function is monotone at every point. The novelty of our work on Dini domains is how to compensate for the lack of such monotone quantities at boundary as well as interior points.

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.

The purpose of this paper is to study the eigenvalues { λ μ,i } i for the Dirichlet Hardy-Leray operator, i.e. ?�Δu+μ|x | ?? u=λu in Ω,u=0 on ?��? where ?��? μ |x | 2 is the Hardy-Leray operator with μ?��? (N?? ) 2 4 and Ω is a smooth bounded domain with 0?��?. We provide lower bounds of { λ μ,i } i together with the Li-Yau's one for μ>??(N?? ) 2 4 and Karachalio's one for μ?�[??(N?? ) 2 4 ,0) . Secondly, we obtain Cheng-Yang's type upper bounds for λ μ,k . Finally, we get the Weyl's limit of eigenvalues which is independent of the potential's parameter μ . This interesting phenomena indicates that the inverse-square potential does not play an essential role for the asymptotic behavior of the spectral of the problem considered.

In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this paper we study a related steady version of this problem where a steadily stirred fluid is uniformly heated in the interior and cooled on the boundary. For a given large Péclet number, how should one stir to minimize some norm of the temperature? This version of the problem was previously studied by Marcotte, Doering et\ al.\ (SIAM Appl.\ Math '18) in a disk, where the authors used matched asymptotics to show that when the Péclet number, Pe , is sufficiently large one can stir the fluid in a manner that ensures the total heat is O(1/Pe) . In this paper we confirm their results with rigorous proofs, and also provide an almost matching lower bound. For simplicity, we work on the infinite strip instead of the unit disk and the proof uses probabilistic techniques.

We develop a global bifurcation theory for two classes of nonlinear elastic materials. It is supposed that they are subjected to anti-plane shear deformation and occupy an infinite cylinder in the reference configuration. Curves of solutions to the corresponding elastostatic problem are constructed using analytic global bifurcation theory. The curve associated with first class is shown to exhibit broadening behavior, while for the second we find that the governing equation undergoes a loss ellipticity in the limit. A sequence of solutions undergoes broadening when their effective supports grow without bound. This phenomena has received considerable attention in the context of solitary water waves; it has been predicted numerically, yet it remains to be proven rigorously. The breakdown of ellipticity is related to cracks and instability making it an important aspect of the theory of failure mechanics.

We show that the singular set Σ in the classical obstacle problem can be locally covered by a C ??hypersurface, up to an "exceptional" set E , which has Hausdorff dimension at most n?? (countable, in the n=2 case). Outside this exceptional set, the solution admits a polynomial expansion of arbitrarily large order. We also prove that Σ?�E is extremely unstable with respect to monotone perturbations of the boundary datum. We apply this result to the planar Hele-Shaw flow, showing that the free boundary can have singular points for at most countable many times.

We describe a functional framework suitable to the analysis of the Cahn-Hilliard equation on an evolving surface whose evolution is assumed to be given \textit{a priori}. The model is derived from balance laws for an order parameter with an associated Cahn-Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular nonlinearities -- the thermodynamically relevant logarithmic potential and a double obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution, and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

In the present paper, we consider an elliptic divergence form operator in the half-space and prove that its Green function is almost affine, or more precisely, that the normalized difference between the Green function and a suitable affine function at every scale satisfies a Carleson measure estimate, provided that the oscillations of the coefficients satisfy the traditional quadratic Carleson condition. The results are sharp, and in particular, it is demonstrated that the class of the operators considered in the paper cannot be improved.

Ivrii's conjecture asserts that the Cauchy problem is C ??well-posed for any lower order term if every critical point of the principal symbol is effectively hyperbolic. Effectively hyperbolic critical point is at most triple characteristic. If every characteristic is at most double this conjecture has been proved in 1980's. In this paper we prove the conjecture for the remaining cases, that is for operators with triple effectively hyperbolic characteristics.

The initial boundary value problem for a nonlinear system of equations modeling the chevron patterns is studied in one and two spatial dimensions. The existence of an exponential attractor and the stabilization of the zero steady state solution through application of a finite-dimensional feedback control is proved in two spatial dimensions. The stabilization of an arbitrary fixed solution is shown in one spatial dimension along with relevant numerical results.