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In an earlier paper, [1] , the authors treated a family of Allen–Cahn model problems for which 0 and 1 are solutions and further solutions were found that are near 1 on a prescribed set, T + Ω , where T ⊂ Z n , and near 0 on ( Z n ∖ T ) + Ω . Here Ω ⊂ ( 0 , 1 ) n . In this paper, a more general class of potentials is treated for which the pair, { 0 , 1 } , is replaced by Z and the existence of a far richer structure of shadowing solutions, including unbounded ones, is established. [ABSTRACT FROM AUTHOR]

The Hunter–Saxton equation determines a flow of conservative solutions taking values in the space H 1 ( R + ) . However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural H 1 distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter–Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H 1 ( R + ) . [ABSTRACT FROM AUTHOR]

In this paper, we prove explicit lower bounds for the cost of fast boundary controls for a class of linear equations of parabolic or dispersive type involving the spectral fractional Laplace operator. We notably deduce the following striking result: in the case of the heat equation controlled on the boundary, Miller's conjecture formulated in Miller (2004) [16] is not verified. Moreover, we also give a new lower bound for the minimal time needed to ensure the uniform controllability of the one-dimensional convection–diffusion equation with negative speed controlled on the left boundary, proving that the conjecture formulated in Coron and Guerrero (2005) [2] concerning this problem is also not verified at least for negative speeds. The proof is based on complex analysis, and more precisely on a representation formula for entire functions of exponential type, and is quite related to the moment method . [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the existence of segregated vector solutions for where Ω is a bounded or unbounded domain in with , is a small parameter, is an integer, ( ) is a potential function, ( ) is constant and ( ) is coupling constant. This system describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photorefractive in optics and Bose–Einstein condensates. For , which corresponds to the synchronization case for the above system with constant potentials, we prove that the system has multiple positive vector solutions, whose components may have spikes clustering at the same point as , but the distance between them divided by ε will go to infinity. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with the partial regularity of the stationary solution to semilinear elliptic equations. For almost all supercritical exponents, we improve the upper bounds of the Hausdorff dimension of the singular set, which are derived by Pacard, to the optimal ones. [Copyright &y& Elsevier]

Abstract: In this paper, we give sharp estimates of the smallest principal curvature of level sets of n-dimensional p-harmonic functions which extends the result of 2-dimensional minimal surface case due to Longinetti [Longinetti, On minimal surfaces bounded by two convex curves in parallel planes, J. Differential Equations 67 (3) (1987) 344–358]. More precisely, we prove that the function is a convex function with respect to the layer parameter of the level sets for all and . [Copyright &y& Elsevier]

Abstract: We consider nonlinear elliptic partial differential equations for quasilinear operators of the form subject to fully nonlinear boundary conditions involving boundary operators of the form, for each , The main goal of this paper is to give, under suitable assumptions on and , an explicit estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators which are, possibly, of the same order as . As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales. [Copyright &y& Elsevier]