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It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [13] . The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [10] . In this paper, we confirm that the other two are also area-minimizing. In fact, we show that every Lawson–Osserman cone of type ( n , p , 2 ) constructed in [26] is area-minimizing. [ABSTRACT FROM AUTHOR]

In this paper we consider finite energy ℓ -equivariant wave maps from R t , x 1 + 3 \ ( R × B ( 0 , 1 ) ) → S 3 with a Dirichlet boundary condition at r = 1 , and for all ℓ ∈ N . Each such ℓ -equivariant wave map has a fixed integer-valued topological degree, and in each degree class there is a unique harmonic map, which minimizes the energy for maps of the same degree. We prove that an arbitrary ℓ -equivariant exterior wave map with finite energy scatters to the unique harmonic map in its degree class, i.e., soliton resolution. This extends the recent results of the first, second, and fourth authors on the 1-equivariant equation to higher equivariance classes, and thus completely resolves a conjecture of Bizoń, Chmaj and Maliborski, who observed this asymptotic behavior numerically. The proof relies crucially on exterior energy estimates for the free radial wave equation in dimension d = 2 ℓ + 3 , which are established in the companion paper [13] . [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of maps minimising the energy ∫ D ( | ∇ u | 2 + 2 | u | ) d x , which, due to Lipschitz character of the integrand, gives rise to the singular Euler equations Δ u = u | u | χ { | u | > 0 } , u = ( u 1 , ⋯ , u m ) . Our primary goal in this paper is to set up a road map for future developments of the theory related to such energy minimising maps. Our results here concern regularity of the solution as well as that of the free boundary. They are achieved by using monotonicity formulas and epiperimetric inequalities, in combination with geometric analysis. [ABSTRACT FROM AUTHOR]

This paper investigates realisations of elliptic differential operators of general order on manifolds with boundary following the approach of Bär-Ballmann to first order elliptic operators. The space of possible boundary values of elements in the maximal domain is described as a Hilbert space densely sandwiched between two mixed order Sobolev spaces. The description uses Calderón projectors which, in the first order case, is equivalent to results of Bär-Bandara using spectral projectors of an adapted boundary operator. Boundary conditions that induce Fredholm as well as regular realisations, and those that admit higher order regularity, are characterised. In addition, results concerning spectral theory, homotopy invariance of the Fredholm index, and well-posedness for higher order elliptic boundary value problems are proven. [ABSTRACT FROM AUTHOR]

In this paper we prove the local Lipschitz regularity of the minimizers of the two-phase Bernoulli type free boundary problem arising from the minimization of the functional J ( u ) : = ∫ Ω | ∇ u | p + λ + p χ { u > 0 } + λ − p χ { u ≤ 0 } , 1 < p < ∞ . Here Ω ⊂ R N is a bounded smooth domain and λ ± are positive constants such that λ + p − λ − p > 0 . Furthermore, we show that for p > 1 the free boundary has locally finite perimeter and the set of non-smooth points of the free boundary is of zero ( N − 1 ) -dimensional Hausdorff measure. For this, our approach is new even for the classical case p = 2 . [ABSTRACT FROM AUTHOR]

In this paper, we study the Heisenberg uniqueness pairs corresponding to a finite number of parallel lines Γ. We give a necessary condition and a sufficient condition for a subset Λ of R 2 so that ( Γ , Λ ) becomes a HUP. [ABSTRACT FROM AUTHOR]

The problem of determining the set of possible eigenvalues of 3 Hermitian matrices that sum up to zero is known as the Horn problem. The answer is a polyhedral cone, which, following Knutson and Tao, can be described as the projection of a simpler cone in the space of triangular tableaux (or hives ) to the boundary nodes of the tableau. In this paper, we introduce a combinatorial problem, defined in terms of certain weighted planar graphs, giving rise to exactly the same polyhedral cone. In our framework, the values at the inner nodes of the triangular tableaux receive a natural interpretation. Other problems of linear algebra fit into the same scheme, among them the Gelfand–Zeitlin problem. Our approach is motivated by the works of Fomin and Zelevinsky on total positivity and by the ideas of tropicalization. [ABSTRACT FROM AUTHOR]

We are concerned with the stability of multidimensional (M-D) transonic shocks in steady supersonic flow past multidimensional wedges. One of our motivations is that the global stability issue for the M-D case is much more sensitive than that for the 2-D case, which requires more careful rigorous mathematical analysis. In this paper, we develop a nonlinear approach and employ it to establish the stability of weak shock solutions containing a transonic shock-front for potential flow with respect to the M-D perturbation of the wedge boundary in appropriate function spaces. To achieve this, we first formulate the stability problem as a free boundary problem for nonlinear elliptic equations. Then we introduce the partial hodograph transformation to reduce the free boundary problem into a fixed boundary value problem near a background solution with fully nonlinear boundary conditions for second-order nonlinear elliptic equations in an unbounded domain. To solve this reduced problem, we linearize the nonlinear problem on the background shock solution and then, after solving this linearized elliptic problem, develop a nonlinear iteration scheme that is proved to be contractive. [ABSTRACT FROM AUTHOR]

This paper discovers two relationships between the [ 1 , n ) ∋ p -capacity and the surface-area via the Lebesgue volume and the Willmore energy in R n respectively, whence showing: if Ω ⊂ R n is a convex, compact, smooth set with its interior Ω ∘ ≠ ∅ and the mean curvature H ( ∂ Ω , ⋅ ) > 0 of its boundary ∂Ω then ( n ( p − 1 ) p ( n − 1 ) ) p − 1 ≤ ( cap p ( Ω ) ( p − 1 n − p ) 1 − p σ n − 1 ) ( area ( ∂ Ω ) σ n − 1 ) n − p n − 1 ≤ ( ∫ ∂ Ω ( H ( ∂ Ω , ⋅ ) ) n − 1 d σ ( ⋅ ) σ n − 1 ) p − 1 n − 1 and hence c a p 1 ( Ω ) area ( ∂ Ω ) = 1 ≤ ∫ ∂ Ω ( H ( ∂ Ω , ⋅ ) ) n − 1 d σ ( ⋅ ) σ n − 1 . Of particular interest is that if p = 2 < 3 = n in the last but one inequality then cap 2 ( Ω ) ≥ 2 − 1 3 π area ( ∂ Ω ) where 2 − 1 3 π is smaller than 2 2 2 / π (conjectured by Pólya–Szegö in [25, (2)] ) while bigger than 2 2 / π (obtained by Pólya–Szegö in [27, p. 165(4)] ). [ABSTRACT FROM AUTHOR]

In this paper we solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in R n . The main results apply, in particular, to subequations with a Riesz characteristic p ≥ 2 . It is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel Θ j K p ( x − x j ) where K p ( x ) = { − 1 | x | p − 2 for 2 < p < ∞ , log ⁡ | x | if p = 2 . at any prescribed finite set of points { x 1 , . . . , x k } in the domain and any finite set of positive real numbers Θ 1 , . . . , Θ k . This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as Θ j | x − x j | 2 − p for 1 ≤ p < 2 . The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong–Sirakov–Smart (in the uniformly elliptic case). [ABSTRACT FROM AUTHOR]

In conventional Differential Geometry one studies manifolds, locally modelled on R n , manifolds with boundary, locally modelled on [ 0 , ∞ ) × R n − 1 , and manifolds with corners, locally modelled on [ 0 , ∞ ) k × R n − k . They form categories Man ⊂ M a n b ⊂ M a n c . Manifolds with corners X have boundaries ∂ X , also manifolds with corners, with dim ∂ X = dim X − 1 . We introduce a new notion of manifolds with generalized corners , or manifolds with g-corners , extending manifolds with corners, which form a category M a n gc with Man ⊂ M a n b ⊂ M a n c ⊂ M a n gc . Manifolds with g-corners are locally modelled on X P = Hom Mon ( P , [ 0 , ∞ ) ) for P a weakly toric monoid, where X P ≅ [ 0 , ∞ ) k × R n − k for P = N k × Z n − k . Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries ∂ X . In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in M a n gc exist under much weaker conditions than in M a n c . This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of J -holomorphic curves can be manifolds or Kuranishi spaces with g-corners rather than ordinary corners. Our manifolds with g-corners are related to the ‘interior binomial varieties’ of Kottke and Melrose [20] , and the ‘positive log differentiable spaces’ of Gillam and Molcho [6] . [ABSTRACT FROM AUTHOR]

This paper proposes a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of d -rectifiable closed subsets of R n : following the previous work [13] , the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than ( d − 1 ) . [ABSTRACT FROM AUTHOR]

The conformal Riemann mapping of the unit disk onto a simply-connected domain W is a central object of study in classical Complex Analysis. The first complete proof of the Riemann Mapping Theorem given by P. Koebe in 1912 is constructive, and theoretical aspects of computing the Riemann map have been extensively studied since. Carathéodory Theory describes the boundary extension of the Riemann map. In this paper we develop its constructive version with explicit complexity bounds. [ABSTRACT FROM AUTHOR]

Abstract: This paper is on further development of discrete complex analysis introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We consider a graph lying in the complex plane and having quadrilateral faces. A function on the vertices is called discrete analytic, if for each face the difference quotients along the two diagonals are equal. We prove that the Dirichlet boundary value problem for the real part of a discrete analytic function has a unique solution. In the case when each face has orthogonal diagonals we prove that this solution uniformly converges to a harmonic function in the scaling limit. This solves a problem of S. Smirnov from 2010. This was proved earlier by R. Courant–K. Friedrichs–H. Lewy and L. Lusternik for square lattices, by D. Chelkak–S. Smirnov and implicitly by P.G. Ciarlet–P.-A. Raviart for rhombic lattices. In particular, our result implies uniform convergence of the finite element method on Delaunay triangulations. This solves a problem of A. Bobenko from 2011. The methodology is based on energy estimates inspired by alternating-current network theory. [Copyright &y& Elsevier]

Abstract: We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier–Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data. [Copyright &y& Elsevier]

Abstract: In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second result, more technical, but also connected to the first one, is an extension of the well known Pietsch Domination Theorem. The last decade witnessed the birth of different families of Pietsch Domination-type results and some attempts of unification. Our result, that we call “full general Pietsch Domination Theorem” is potentially a definitive Pietsch Domination Theorem which unifies the previous versions and delimits what can be proved in this line. The connections to the recent notion of weighted summability are traced. [Copyright &y& Elsevier]

Abstract: In this paper we study the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator and we prove, in particular, that Lipschitz free boundaries are -smooth for some . As part of our argument, and which is of independent interest, we establish a Hopf boundary type principle for non-negative p-harmonic functions vanishing on a portion of the boundary of a Lipschitz domain. [ABSTRACT FROM AUTHOR]

Abstract: In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem: where Ω is a bounded smooth domain of , and is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems. [Copyright &y& Elsevier]

Abstract: For the parabolic obstacle-problem-like equation where and are positive Lipschitz functions, we prove in arbitrary finite dimension that the free boundary is in a neighborhood of each “branch point” the union of two Lipschitz graphs that are continuously differentiable with respect to the space variables. The result extends the elliptic paper [Henrik Shahgholian, Nina Uraltseva, Georg S. Weiss, The two-phase membrane problem—regularity in higher dimensions, Int. Math. Res. Not. (8) (2007)] to the parabolic case. There are substantial difficulties in the parabolic case due to the fact that the time derivative of the solution is in general not a continuous function. Our result is optimal in the sense that the graphs are in general not better than Lipschitz, as shown by a counter-example. [Copyright &y& Elsevier]

Abstract: We generalize the reduction theorem for 0-parameter solutions of a traditional (i.e., second order) Painlevé equation with a large parameter to those of some higher order Painlevé equation, that is, each member of the Painlevé hierarchies or II-2). Thus the scope of applicability of the reduction theorem in [KT1] has been substantially enlarged; only six equations were covered by our previous result, while Theorem 3.2.1 of this paper applies to infinitely many equations. [Copyright &y& Elsevier]

The formation of thinning filaments is commonly observed previously to the break-up of a very viscous jet. This paper shows that a fluid under capillary forces cannot break-up through the uniform collapse of a filament. [Copyright &y& Elsevier]

In this paper we establish a global existence of weak solutions to the two-dimensional Prandtl''s system for unsteady boundary layers in the class considered by Oleinik (J. Appl. Math. Mech. 30 (1966) 951) provided that the pressure is favourable. This generalizes the local well-posedness results due to Oleinik (1966; Mathematical Models in Boundary Layer Theory, Chapman & Hall, London, 1999). For the proof, we introduce a viscous splitting method so that the asymptotic behaviour of the solution near the boundary can be estimated more accurately by methods applicable to the degenerate parabolic equations. [Copyright &y& Elsevier]

This paper studies several aspects of asymptotically hyperbolic (AH) Einstein metrics, mostly on 4-manifolds. We prove boundary regularity (at infinity) for such metrics and establish uniqueness under natural conditions on the boundary data. By examination of explicit black hole metrics, it is shown that neither uniqueness nor finiteness holds in general for AH Einstein metrics with a prescribed conformal infinity. We then describe natural conditions which are sufficient to ensure finiteness. [Copyright &y& Elsevier]