Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

We establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C^1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C^1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C^1,α, provided that the hyperbolic phase is close in C^1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C^2,α, the free boundary is C^2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.     

Schauder Estimates for the Single Layer Potential in Hydrodynamics

Let Ω be an open set in ?^N(N ? 3), with compact boundary ?Ω of type C^1,α(?(0,1)). We show that the single layer potential Ef, related to the stationary Stokes system on Ω, belongs to C^1,α(?Ω)^N, provided the source density f belongs to C^α(?Ω)^N. In addition, we prove a related estimate of the function E(f)_?Ω and its tangential derivatives.     

An existence and uniqueness result of a nonlinear two-dimensional elliptic boundary value problem

We consider a boundary value problem for the generalized two-dimensional flow equation Δφ = Δφ · h for h a C^α vector field, where the speed is prescribed on a part of the boundary. By using Bers theory combined with elliptic operator theory in nonsmooth domains, we show existence and uniqueness of a C^2,α solution with nonvanishing gradient, and we find positive lower and upper bounds for |Δφ| along with C^2,α estimates of φ, in terms of the C^α and L^∞ norms of h. ©1995 John Wiley & Sons, Inc.     

Estimating ∇u in terms of div u,curl u,either (ν, u) or ν×u and the topology

In the present paper we prove C^α-estimates for ∇ u using components of boundary values of u , div u , curl u and quantities given by components of boundary values of u as well as boundary values of elements belonging to de Rhams cohomology modules. The vector field u is defined on a bounded set G¯⊂ℝ³, meanwhile the cohomology group will be defined with regard to ℝ³−G. Our inequalities turn out to be a priori estimates concerning well-known boundary value problems for vector fields. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Regularity of free boundaries of two‐phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are C1,α

In this paper, we study an extension of a C^1,α regularity theory developed by L. Caffarelli in [2] to some fully nonlinear elliptic equations of second order. In fact, we investigate a two‐phase free boundary problem in which a fully nonlinear elliptic equation of second order is verified by the solution in the positive and the negative domains. Assuming the free boundary is locally a Lipschitz graph, we have established the C^1,α regularity of the free boundary. © 2000 John Wiley & Sons, Inc.     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Partial regularity of solutions of fully nonlinear,uniformly elliptic equations

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C^2,α on the complement of a closed set of Hausdorff dimension at most ? less than the dimension. The equation is assumed to be C¹, and the constant ? > 0 depends only on the dimension and the ellipticity constants. The argument combines the W^2,? estimates of Lin with a result of Savin on the C^2,α regularity of viscosity solutions that are close to quadratic polynomials. © 2012 Wiley Periodicals, Inc.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

Regularity Near a Contact Point for Flame Propagation

In this paper we study a free boundary problem, arising from a model for the propagation of laminar flames. Consider a cylindrical region S in ? ⁿ , and the following free boundary problem with Dirichlet data on ? S: u _t  = Δ u in {u > 0} ∩ S, |? u|=1 on ? {u > 0} ∩ S and u = 0 on ? S. We show that if there is a contact point of the free boundary {u = 0, |? u|=1} with ? S, then the free boundary approaches ? S tangentially and it turns out to be a graph of C ^{1+α, α} function near the contact point. In particular, the space normal is Hölder continuous.     

Regularity in a one-phase free boundary problem for the fractional Laplacian

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are C^1,α$C^{1,\alpha }$. We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.     

On the traction problem of classical elastostatics

We prove that the traction problem of homogeneous and isotropic elastostatics has a unique classical solution in bounded and exterior domains of class C^2,α for continuous boundary data.     

A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Given a domain Ω of class C ^k,1, k ∈ ℕ, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of in the sense that (∂/∂x _n )α(x′, 0) = − N(x′) and that still is of class C ^k,1. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C ^k,1. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.     

Calculus of variations and descriptive set theory

If X is a locally compact Polish space, then LSC(X, ?) denotes the compact Polish space of lower semi‐continuous real‐valued functions on X equipped with the topology of epi‐convergence. Our purpose in this article is to prove the following: if –∞ < α < β < ∞ and –∞ < a < b < ∞, while r ∈ ? \ {0}, then the set CV of all f ∈ LSC([α, β ] × [a, b ] × ?, ?) for which there is u ∈ C^r ([α, β ], [a, b ]) such that for any v ∈ C^r ([α, β ], [a, b ]) we have that ∫_α^β f (x, u (x), v ′(x))dx ≥ ∫_α^β f (x, v (x), v (x))dx is not Borel (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

REGULARITY OF FREE BOUNDARIES OF TWO-PHASE PROBLEMS FOR FULLY NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER. II. FLAT FREE BOUNDARIES ARE LIPSCHITZ

ABSTRACT

In this second paper, we continue our study on the regularity of free boundaries for some fully nonlinear elliptic equations. Our result is if the free boundary is trapped in a sufficiently narrow strip formed by two Lipschitz graphs, then it is also a Lipschitz graph. Combining with the results in Part 1 (see Ref. [Wang]), the free boundary is C ^1,α.     

A Frequency Function and Singular Set Bounds for Branched Minimal Immersions

We establish a frequency function monotonicity formula for two‐valued C^1,α solutions to the minimal surface system on n‐dimensional domains. We also establish the sharp regularity result that such solutions are of class C^{1, 1/2}, and that their branch sets, if nonempty, have Hausdorff dimension equal to n‐2.© 2016 Wiley Periodicals, Inc.     

The boundary trace and generalized boundary value problem for semilinear elliptic equations with coercive absorption

In the first part of the paper we establish the existence of a boundary trace for positive solutions of the equation ?Δu + g(x, u) = 0 in a smooth domain Ω ? ?^N, for a general class of positive nonlinearities. This class includes every space independent, monotone increasing g which satisfies the Keller‐Osserman condition as well as degenerate nonlinearities g_α,q of the form g_α,q (x, u) = d(x, ?Ω)^α |u|^q?1 u, with α > ?2 and q > 1. The boundary trace is given by a positive regular Borel measure which may blow up on compact sets. In the second part we concentrate on the family of nonlinearities {g_α,q}, determine the critical value of the exponent q (for fixed α > ?2) and discuss (a) positive solutions with an isolated singularity, for subcritical nonlinearities and (b) the boundary value problem for ?Δu + g_α,q (x, u) = 0 with boundary data given by a positive regular Borel measure (possibly unbounded). We show that, in the subcritical case, the problem possesses a unique solution for every such measure. © 2003 Wiley Periodicals, Inc.     

Classes of interval graphs under expanding length restrictions

Let C(α) denote the finite interval graphs representable as intersection graphs of closed real intervals with lengths in [1, α]. The points of increase for C are the rational α ≥ 1. The set D(α) = [∩_β>αC(β)]\C(α) of graphs that appear as soon as we go past α is characterized up to isomorphism on the basis of finite sets E(α) of irreducible graphs for each rational α. With α = p/q and p and q relatively prime, ∣E(α)∣ is computed for all (p,q) with q ? 2 and p = q + 1. When q = 1, E(p) contains only the bipartite star K¹, p_+2. A lowr bound on ∣E(α)∣ is given for all rational α.     

Torsion in Boundary Coinvariants and K -Theory for Affine Buildings

Let (G, I, N, S) be an affine topological Tits system, and let Γ be a torsion-free cocompact lattice in G. This article studies the coinvariants H ₀(Γ; C(Ω,Z)), where Ω is the Furstenberg boundary of G. It is shown that the class [1] of the identity function in H ₀(Γ; C(Ω, Z)) has finite order, with explicit bounds for the order. A similar statement applies to the K ₀ group of the boundary crossed product C ^*-algebra C(Ω)Γ. If the Tits system has type ? ₂, exact computations are given, both for the crossed product algebra and for the reduced group C ^*-algebra.     

Identifiability at the boundary for first-order terms

Let Ω be a domain in R ⁿ whose boundary is C ¹ if n≥3 or C ^1,β if n=2. We consider a magnetic Schrödinger operator L _{W , q} in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L _{W , q} . We also consider a steady state heat equation with convection term Δ+2W·? and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.     

On <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">1</Emphasis></Subscript> of a self-product of a curve

We present elements of H¹(C×C, ᵊ6₂) for certain specific curves C. The image of the element under the boundary map arising from the localization sequence of K-theory is the graph of frobenius endomorphism of the reduction of the curve modulo 3.