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We investigate the existence and properties of Lipschitz solutions for some forward-backward parabolic equations in all dimensions. Our main approach to existence is motivated by reformulating such equations into partial differential inclusions and relies on a Baire's category method. In this way, the existence of infinitely many Lipschitz solutions to certain initial-boundary value problem of those equations is guaranteed under a pivotal density condition. Finally, we study two important cases of forward-backward anisotropic diffusion in which the density condition can be realized and therefore the existence results follow together with micro-oscillatory behavior of solutions. The first case is a generalization of the Perona-Malik model in image processing and the other that of Höllig's model related to the Clausius-Duhem inequality in the second law of thermodynamics.2010 Mathematics Subject Classification. 35M13, 35K20, 35D30, 49K20. Key words and phrases. forward-backward parabolic equations, partial differential inclusions, convex integration, Baire's category method, infinitely many Lipschitz solutions. 1 arXiv:1506.05847v1 [math.AP] 18 Jun 2015 Ω u(x, t)dx = Ω u 0 (x)dx ∀t ∈ [0, T ].

Abstract. We prove that for all smooth nonconstant initial data the initialNeumann boundary value problem for the Perona-Malik equation in image processing possesses infinitely many Lipschitz weak solutions on smooth bounded convex domains in all dimensions. Such existence results have not been known except for the one-dimensional problems. Our approach is motivated by reformulating the Perona-Malik equation as a nonhomogeneous partial differential inclusion with linear constraint and uncontrollable components of gradient. We establish a general existence result by a suitable Baire's category method under a pivotal density hypothesis. We finally fulfill this density hypothesis by convex integration based on certain approximations from an explicit formula of lamination convex hull of some matrix set involved.

We make some remarks about rank-one convex and polyconvex functions on the set of all realn×nmatrices that vanish on the subsetKnconsisting of all conformal matrices and grow like a power function at infinity. We prove that every non-negative rank-one convex function that vanishes onKnand grows below a power of degreen/2 must vanish identically. In odd dimensionsn≧ 3, we prove that every non-negative polyconvex function that vanishes onKnmust vanish identically if it grows below a power of degreen; while in even dimensions, such polyconvex functions can exist that also grow like a power of half-dimension degree.

Abstract. Given a number L ≥ 1, a weakly L-quasiregular map on a domain Ω in space R n is a map u in a Sobolev spaceIn this paper, we study the problem concerning linear boundary values of weakly L-quasiregular mappings in space R n with dimension n ≥ 3. It turns out this problem depends on the power p of the underlying Sobolev space. For p not too far below the dimension n we show that a weakly quasiregular map in W 1,p (Ω; R n ) can only assume a quasiregular linear boundary value; however, for all L ≥ 1 and 1 ≤ p < nL L+1 , we prove a rather surprising existence result that every linear map can be the boundary value of a weakly L-quasiregular map in W 1,p (Ω; R n ). Classification (1991):30C65, 30C70, 35F30, 49J30 Mathematics Subject

We present a new method for micromagnetics based on replacing the nonlocal total energy of magnetizations by a new local energy for divergence-free fields and then studying the dual Legendre functional of this new energy restricted on gradient fields. We establish a Fencheltype duality principle relevant to the minimization for these problems. The dual functional may be written as a convex integral functional of gradients, and its minimization problem will be solved by standard minimization procedures in the calculus of variations. Special emphasis is placed on the analysis of existence/nonexistence, depending on the applied field and the physical domain. In particular, we describe a precise procedure to check the existence of magnetization of minimal energy for ellipsoid domains.

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with major application in image processing. In this paper we study the Perona-Malik type equation and show that, in all dimensions, there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for any smooth nonconstant radially symmetric initial data. Our approach is to reformulate the n-dimensional equation into a one-dimensional equation, to convert the one-dimensional problem into a differential inclusion problem, and to apply a Baire's category method to generate infinitely many solutions.

Abstract. We adopt the idea of Baire's category method as presented in a series of papers by Dacorogna and Marcellini to study the boundary value problem for quasiregular mappings in space. Our main result is to prove that for any > 0 and any piece-wise affine mapThe theorems of Dacorogna and Marcellini do not directly apply to our result since the involved sets are unbounded. Our proof is elementary and does not require any notion of polyconvexity, quasiconvexity or rank-one convexity in the vectorial calculus of variations, as required in the papers by the quoted authors.