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  • 1.
    Auscher, Pascal
    et al.
    Université Paris-Sud, France.
    Rosén, Andreas
    Chalmers University, Sweden.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Boundary value problems for degenerate elliptic equations and systems2015In: Annales Scientifiques de l'Ecole Normale Supérieure, ISSN 0012-9593, E-ISSN 1873-2151, Vol. 48, no 4, p. 951-1000Article in journal (Refereed)
  • 2.
    Carbery, Anthony
    et al.
    Edinburgh University, UK.
    Maz'ya, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Mitrea, Marius
    University of Missouri, USA.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The integrability of negative powers of the solution of the Saint Venant problem2014In: Annali della Scuola Normale Superiore di Pisa (Classe Scienze), Serie V, ISSN 0391-173X, E-ISSN 2036-2145, Vol. XIII, no 2, p. 465-531Article in journal (Refereed)
    Abstract [en]

    We initiate the study of the finiteness condition∫ Ω u(x) −β dx≤C(Ω,β)<+∞ whereΩ⊆R n is an open set and u is the solution of the Saint Venant problem Δu=−1 in Ω , u=0 on ∂Ω . The central issue which we address is that of determining the range of values of the parameter β>0 for which the aforementioned condition holds under various hypotheses on the smoothness of Ω and demands on the nature of the constant C(Ω,β) . Classes of domains for which our analysis applies include bounded piecewise C 1 domains in R n , n≥2 , with conical singularities (in particular polygonal domains in the plane), polyhedra in R 3 , and bounded domains which are locally of classC 2 and which have (finitely many) outwardly pointing cusps. For example, we show that if u N is the solution of the Saint Venant problem in the regular polygon Ω N with N sides circumscribed by the unit disc in the plane, then for each β∈(0,1) the following asymptotic formula holds: % {eqnarray*} \int_{\Omega_N}u_N(x)^{-\beta}\,dx=\frac{4^\beta\pi}{1-\beta} +{\mathcal{O}}(N^{\beta-1})\quad{as}\,\,N\to\infty. {eqnarray*} % One of the original motivations for addressing the aforementioned issues was the study of sublevel set estimates for functions v satisfying v(0)=0 , ∇v(0)=0 and Δv≥c>0 .

  • 3.
    Dindos, Martin
    et al.
    School of Mathematics Edinburgh University Mayfield Road Edinburgh, EH9 3JZ, UK.
    Pipher, Jill
    Brown University Mathematics Department Providence, RI 02912, USA.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Boundary value problems for second order elliptic operators satisfying a Carleson condition2017In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 70, no 7, p. 1316-1365Article in journal (Refereed)
    Abstract [en]

    Let be a Lipschitz domain in Rn n ≥ 2, and L = divA∇· be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in H1,p(@) and of the Neumann problem with Lp(@) data for the operator L on Lipschitz domains with small Lipschitz con- stant. We allow the coefficients of the operator L to be rough obeying a certain Carleson condition with small norm. These results complete the results of [7] where the Lp(@) Dirichlet problem was considered under the same assumptions and [8] where the regularity and Neumann problems were considered on two dimensional domains.

  • 4.
    Ghosh, Arpan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergei A.
    St Petersburg State University, Russia; St Petersburg State Polytech University, Russia.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A Two Dimensional Model of the Thin Laminar Wall of a Curvilinear Flexible Pipe2018Report (Other academic)
    Abstract [en]

    We present a two dimensional model describing the elastic behaviour of the wall of a curved pipe to model blood vessels in particular. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the vessel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with the surrounding material and the fluid flowing inside into account and is obtained via a dimension reduction procedure. The curvature and twist of the vessel axis as well as the anisotropy of the laminate wallpresent the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of vessels and their walls are supplied with explicit systems of dierential equations at the end.

  • 5.
    Ghosh, Arpan
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kozlov, Vladimir
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nazarov, Sergey
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. St Petersburg State Univ, Russia; RAS, Russia.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    A TWO-DIMENSIONAL MODEL OF THE THIN LAMINAR WALL OF A CURVILINEAR FLEXIBLE PIPE2018In: Quarterly Journal of Mechanics and Applied Mathematics, ISSN 0033-5614, E-ISSN 1464-3855, Vol. 71, no 3, p. 349-367Article in journal (Refereed)
    Abstract [en]

    We present a two-dimensional model describing the elastic behaviour of the wall of a curved flexible pipe. The wall has a laminate structure consisting of several anisotropic layers of varying thickness and is assumed to be much smaller in thickness than the radius of the channel which itself is allowed to vary. Our two-dimensional model takes the interaction of the wall with any surrounding or supporting material and the fluid flow into account and is obtained via a dimension reduction procedure. The curvature and twist of the pipes axis as well as the anisotropy of the laminate wall present the main challenges in applying the dimension reduction procedure so plenty of examples of canonical shapes of pipes and their walls are supplied with explicit systems of differential equations at the end.

  • 6.
    Michalowski, Nicholas
    et al.
    New Mexico State University, Las Cruces, NM, USA .
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Staubach, Wolfgang
    Uppsala University, Sweden .
    Multilinear pseudodifferential operators beyond Calderón–Zygmund theory2014In: Journal of Mathematical Analysis and Applications, ISSN 0022-247X, E-ISSN 1096-0813, Vol. 414, no 1, p. 149-165Article in journal (Refereed)
    Abstract [en]

    We consider two types of multilinear pseudodifferential operators. First, we prove the boundedness of multilinear pseudodifferential operators with symbols which are only measurable in the spatial variables in Lebesgue spaces. These results generalise earlier work of the present authors concerning linear pseudo-pseudodifferential operators. Secondly, we investigate the boundedness of bilinear pseudodifferential operators with symbols in the Hormander S-p,delta(m) classes. These results are new in the case p less than 1, that is, outwith the scope of multilinear Calderon-Zygmund theory.

  • 7.
    Rodríguez-López, Salvador
    et al.
    Uppsala University, Sweden.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Staubach, Wolfgang
    Uppsala University, Sweden.
    A Seeger-Sogge-Stein theorem for bilinear Fourier integral operators2014In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 264, p. 1-54Article in journal (Refereed)
    Abstract [en]

    We establish the regularity of bilinear Fourier integral operators with bilinear amplitudes in

    and non-degenerate phase functions, from Lp×Lq→Lr under the assumptions that

     and . This is a bilinear version of the classical theorem  of Seeger–Sogge–Stein concerning the Lp boundedness of linear Fourier integral operators. Moreover, our result goes beyond the aforementioned theorem in that it also includes the case of quasi-Banach target spaces.

  • 8.
    Rodríguez-López, Salvador
    et al.
    Uppsala universitet.
    Rule, David
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Staubach, Wolfgang
    Uppsala universitet.
    On the boundedness of certain bilinear oscillatory integral operators2015In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 367, no 10, p. 6971-6995Article in journal (Refereed)
    Abstract [en]

    We prove the global L2 × L2 → L1 boundedness of bilinear oscillatory integral operators with amplitudes satisfying a Hörmander type condition and phases satisfying appropriate growth as well as the strong non-degeneracy conditions. This is an extension of the corresponding result of R. Coifman and Y. Meyer for bilinear pseudo-differential operators, to the case of oscillatory integral operators.

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