96 research outputs found

    Characterization of function spaces via low regularity mollifiers

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    Smoothness of a function f:RnRf:{\mathbb R}^n\to {\mathbb R} can be measured in terms of the rate of convergence of fρεf\ast\rho_\varepsilon to ff, where ρ\rho is an appropriate mollifier. In the framework of fractional Sobolev spaces, we characterize the "appropriate" mollifiers. We also obtain sufficient conditions, close to being necessary, which ensure that ρ\rho is adapted to a given scale of spaces. Finally, we examine in detail the case where ρ\rho is a characteristic function

    Asymptotic behavior of critical points of an energy involving a loop-well potential

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    We describe the asymptotic behavior of critical points of Ω[(1/2)u2+W(u)/ε2]\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2] when ε0\varepsilon\to 0. Here, WW is a Ginzburg-Landau type potential, vanishing on a simple closed curve Γ\Gamma. Unlike the case of the standard Ginzburg-Landau potential W(u)=(1u2)2/4W(u)=(1-|u|^2)^2/4, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry on WW or Γ\Gamma. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest

    Density in Ws,p(Ω;N)W^{s,p}(\Omega ; N)

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    Let Ω\Omega be a smooth bounded domain in Rn{\mathbb R}^n, 0\textless{}s\textless{}\infty and 1\le p\textless{}\infty. We prove that C(Ω;S1)C^\infty(\overline\Omega\, ; {\mathbb S}^1) is dense in Ws,p(Ω;S1)W^{s,p}(\Omega ; {\mathbb S}^1) except when 1\le sp\textless{}2 and n2n\ge 2. The main ingredient is a new approximation method for Ws,pW^{s,p}-maps when s\textless{}1. With 0\textless{}s\textless{}1, 1\le p\textless{}\infty and sp\textless{}n, Ω\Omega a ball, and NN a general compact connected manifold, we prove that C(Ω;N)C^\infty(\overline\Omega \, ; N) is dense in Ws,p(Ω;N)W^{s,p}(\Omega \, ; N) if and only if π_[sp](N)=0\pi\_{[sp]}(N)=0. This supplements analogous results obtained by Bethuel when s=1s=1, and by Bousquet, Ponce and Van Schaftingen when s=2,3,s=2,3,\ldots [General domains Ω\Omega have been treated by Hang and Lin when s=1s=1; our approach allows to extend their result to s\textless{}1.] The case where s\textgreater{}1, s∉Ns\not\in{\mathbb N}, is still open.Comment: To appear in J. Funct. Anal. 49

    Existence of critical points with semi-stiff boundary conditions for singular perturbation problems in simply connected planar domains

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    Let Ω\Omega be a smooth bounded simply connected domain in R2\mathbb{R}^2. We investigate the existence of critical points of the energy Eε(u)=1/2Ωu2+1/(4ε2)Ω(1u2)2E_\varepsilon (u)=1/2\int_\Omega |\nabla u|^2+1/(4\varepsilon^2)\int_\Omega (1-|u|^2)^2, where the complex map uu has modulus one and prescribed degree dd on the boundary. Under suitable nondegeneracy assumptions on Ω\Omega, we prove existence of critical points for small ε\varepsilon. More can be said when the prescribed degree equals one. First, we obtain existence of critical points in domains close to a disc. Next, we prove that critical points exist in "most" of the domains

    Superposition with subunitary powers in Sobolev spaces

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    International audienceLet 00<aa<11 and set Φ(t)=ta\Phi (t)=|t|^a, tRt\in {\mathbb R}. We prove that the superposition operator uΦ(u)u\mapsto \Phi (u) maps the Sobolev space W1,p(Rn)W^{1,p}({\mathbb R}^n) into the fractional Sobolev space Wa,p/a(Rn)W^{a,p/a}({\mathbb R}^n). We also investigate the case of more general nonlinearities

    S1{\mathbb S}^1-valued Sobolev maps

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    International audienceWe describe the structure of the space Ws,p(Sn;S1)W^{s,p}({\mathbb S}^n ; {\mathbb S}^1), where 00<ss<\infty and 1p1\le p<\infty. According to the values of ss, pp and nn,maps in Ws,p(Sn;S1)W^{s,p}({\mathbb S}^n ; {\mathbb S}^1) can either be characterised by their phases, or by a couple (singular set, phase). Here are two examples:a) W1/2,6(S3;S1)={eıφ;φW1/2,6+W1,3}W^{1/2,6}({\mathbb S}^3 ; {\mathbb S}^1) = \{e^{\imath\varphi};\, \varphi\in W^{1/2,6} + W^{1,3}\};b) W1/2,3(S2;S1)D×{eıφ;φW1/2,3+W1,3/2}W^{1/2,3}( {\mathbb S}^2 ; {\mathbb S}^1) \approx D \times \{e^{\imath \varphi} ;\, \varphi\in W^{1/2,3} + W^{1,3/2}\}. In the second example, DD is an appropriate set of infinite sums of Dirac masses. The sense of "\approx" will be explained in the paper. The presentation is based on a paper of H.-M. Nguyen (C. R. Acad. Sci. Paris 2008), and on a forthcoming paper of the author

    Low regularity function spaces of N-valued maps are contractible

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    International audienceLet MM be a compact Lipschitz submanifold, possibly with boundary, of Rn{\mathbb R}^n. Let NRkN\subset {\mathbb R}^k be an arbitrary set. Let s0s\ge 0 and 1p<1\le p<\infty be such that sp<1sp<1. Then Ws,p(M;N)W^{s, p}(M ; N) is contractible

    On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to L1L^1 vector fields

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    International audienceBourgain and Brezis (J. Amer. Math. Soc. 2003) established, for maps fLn(Tn)f\in L^n({\mathbb T}^n) with zero average, the existence of a solution YW1,nL\vec{Y}\in W^{1,n}\cap L^\infty of (1) div Y=f\vec{Y}=f. Maz'ya (Contemp. Math. vol. 445) proved that if, in addition fHn/21(Tn)f\in H^{n/2-1}({\mathbb T}^n), then (1) can be solved in Hn/2LH^{n/2}\cap L^\infty. Their arguments are quite different. We present an elementary property of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova (Sobolev spaces in mathematics I, 2009). We discuss higher dimensional analogs of the above results

    Lifting default for S1{\mathbb S}^1-valued maps

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    International audienceLet φC([0,1]N;R)\varphi\in C^\infty ([0,1]^N ; {\mathbb R}) and set g=eıφg=e^{\imath\varphi}. When 0011 and sp1sp\ge 1, or when N2N\ge 2, pp>11 and spsp>11. In this work, we establish the existence of a control in the full range 00<ss<11, p1p\ge 1 and N1N\ge 1. In particular, we do not require that sp1sp\ge 1, as in the previous results

    Decomposition of S1{\mathbb S}^1-valued maps in Sobolev spaces

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    International audienceLet n2n\ge 2, ss>00 and p1p\ge 1 be such that 1sp1\le sp<22. We prove that for each map uWs,p(Sn;S1)u\in W^{s,p}({\mathbb S}^n ; {\mathbb S}^1) one can find some φWs,p(Sn;R)\varphi\in W^{s,p}({\mathbb S}^n ; {\mathbb R}) and some vWsp,1(Sn;S1)v\in W^{sp, 1}({\mathbb S}^n ; {\mathbb S}^1) such that u=eıφvu=e^{\imath\varphi}\, v. This yields a decomposition of uu into a part, eıφe^{\imath\varphi}, that has a lifting in Ws,pW^{s,p}, and a map, vv, "smoother" than uu but which need not have a lifting within Ws,pW^{s,p}. Our result generalizes a previous one of Bourgain and Brezis (J. Amer. Math. Soc. 2003), which corresponds to s=1/2s=1/2 and p=2p=2. As a consequence of the above factorization u=eıφvu=e^{\imath\varphi}\, v, we find an intuitive proof of the existence of the Jacobian JuJ u of maps uWs,p(Sn;S1)u\in W^{s, p}({\mathbb S}^n ; {\mathbb S}^1), result originally due to Bourgain, Brezis and the author (Comm. Pure Appl. Math. 2005). By completing a result of Bousquet (J. Anal. Math. 2007), we characterize the distributions of the form JuJ u
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