1,721,027 research outputs found
On the propagation of flatness for second order hypoelliptic operators
No full textFor a class of hypoelliptic operators with real-analytic coefficients, we provide a criterion ensuring a partial analyticity result. As a consequence, even when the "elliptic" strong unique continuation (i.e. a solution of the homogeneous equation which vanishes of infinite order at a point is zero near such a point) fails, a weaker form of "propagation" of zeroes still holds. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/)
On the singular set for solutions to a class of Hamilton-Jacobi-Bellman equations
No full textGiven the viscosity solution, u, of the Cauchy problem ut + (f(t,x),∇u)+r|A(t,x)∇u| = 0 in ]0,T[× Rn with u(0,x) = g(x), we describe the arc structure of the set of the points of non-differentiability for u. Moreover, we give a result on the propagation of singularities along the generalized characteristics
Some properties of semiconcave functions with general modulus
No full textAbstractThe class of semiconcave functions represents a useful generalization of the one of concave functions. Such an extension can be achieved requiring that a function satisfies a suitable one-sided estimate. In this paper, the structure of the set of points at which a semiconcave function fails to be differentiable—the singular set—is studied. First, we prove some results on the existence of arcs contained on the singular set. Then, we show how these abstract results apply to semiconcave solutions of Hamilton–Jacobi equations
On the regularity of the distance near the boundary of an obstacle
Full text linkWe study the regularity of the Euclidean distance function from a given point-wise target of a n-dimensional vector space in the presence of a compact obstacle bounded by a smooth hypersurface. It is known that such a function is semiconcave with fractional modulus one half. We provide a geometrical explanation of the exponent one half. Furthermore, under a natural (weak) assumption on the position of the point-wise target relatively to the obstacle, we show that there exists a point in the boundary of the obstacle so that no better regularity result holds near such a point. As a consequence of this result, we show that the Euclidean metric cannot be extended to a tubular neighborhood of the obstacle, as a Riemannian metric, keeping the property that the associated distances coincide outside the obstacle
Some Remarks on the Dirichlet Problem for the Degenerate Eikonal Equation
No full textIn a bounded domain, we consider the viscosity solution of the homogeneous Dirichlet problem for the degenerate eikonal equation. We provide some sufficient conditions for the (local) Lipschitz regularity of such a function
An interpolation problem in the Denjoy–Carleman classes
Full text linkInspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on [a,b]subset of R and given an increasing divergent sequence dn of positive integers such that the derivative of order dn of f has a growth of the type Mdn, when can we deduce that f is a function in the Denjoy-Carleman class CM([a,b])C<^>M([a,b])? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence dn is needed
structural Properties of Singularities of Semiconcave Functions
No full textA semiconcave function on an open domain of is a function that can be locally represented as the sum of a concave function plus a smooth one. The local structure of the singular set (non-differentiability points) of such a function is studied in this paper. A new technique is presented to detect singularities that propagate along Lipschitz arcs and, more generally, along sets of higher dimension. This approach is then used to analyze the singular set of the distance function from a closed subset of
Propagation of singularities for solutions of nonlinear first order partial differential equations
No full textFew results are available in the mathematical literature for studying the structure of the singular set of a weak solution u of F (x, u, Du) = 0. This paper provides new techniques to analyse such a set when u is semiconcave and F is a nonlinear convex function with respect to p. The main objective achieved here is a classification of the singularities of u that propagate along Lipschitz arcs. Such a propagation phenomenon is also described by means of a generalized characteristics inclusion
Carleman estimates and boundary observability for a coupled parabolic-hyperbolic system
No full textThis paper studies a problem of boundary observability for a coupled
system of parabolic–hyperbolic type. First, we prove some Carleman esti-
mates with singular weights for the heat and for the wave equations. Then
we combine these results to obtain an observability result for the system.
We conclude with a discussion about operators with constant coefficients
Generation of singularities from the initial datum for Hamilton-Jacobi equations
Full text linkWe study the generation of singularities from the initial datum for a solution of the Cauchy problem for a class of Hamilton-Jacobi equations of evolution. For such equations, we give conditions for the existence of singular generalized characteristics starting at the initial time from a given point of the domain, depending on the properties of the proximal subdifferential of the initial datum in a neighbourhood of that point
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