24,334 research outputs found
Lᵖ boundary value problems for elliptic and parabolic operators
No full textThis thesis is motivated by questions regarding the solvability of the Dirichlet and regularity boundary value problem with boundary data in Lᵖ for elliptic and parabolic operators.
To this purpose, we are going to develop perturbation theory for the Dirichlet problem for elliptic operators and for the regularity problem for parabolic operators. More specifically, we show perturbation results for the Dirichlet problem with two elliptic operators L₀ = div(A₀ ∇•) and L₁ = div(A₁ ∇•) with matrices A₀ and A₁ having antisymmetric parts in the space of bounded mean oscillation assuming a Carleson type condition on the difference A₀-A₁. In addition to that, we show that a parabolic version of this Carleson condition on the difference A₀ - A₁ gives rise to perturbation results for the parabolic regularity problem for two parabolic operators L₀ = θₜ - div(A₀∇•) and L₁ = θₜ - div(A₁∇•). In both settings we are going to show small perturbation and large perturbation results, i.e. results with small or large Carleson norm perturbations.
The elliptic Dirichlet boundary value has two major sufficient conditions that guarantee solvability, which are the DKP condition and the -independence condition. We are going to apply the developed elliptic perturbation theory to improve on the DKP condition and extend the class of operators for which a DKP-type condition yields solvability of the Lᵖ Dirichlet problem.
On the other hand, we also discuss and show an extension of the t-independence condition by introducing a mixed L¹ - L∞-Carleson type condition on the derivative of A in transversal direction. Under this condition we can also show solvability of the Lᵖ Dirichlet problem
BMO solvability and the A∞ condition for second order parabolic operators
Full text linkWe prove that a sharp regularity property (A∞A∞) of parabolic measure for operators in certain time-varying domains is equivalent to a Carleson measure property of bounded solutions. This equivalence was established in the elliptic case by Kenig, Kirchheim, Pipher and Toro, improving an earlier result of Kenig, Dindos and Pipher for solutions with data in BMO. The connection between regularity of the elliptic measure and certain Carleson measure properties of solutions was established in order to study solvability of boundary value problems for non-symmetric divergence form operators (Kenig, Koch, Pipher, and Toro). The extension to the parabolic setting requires an approach to the key estimates of the aforementioned works that primarily exploits the maximum principle. For various classes of parabolic operators ( [24]), this criterion also provides an easier route to establish the solvability of the Dirichlet problem with data in LpLp for some p, and also to quantify these results in several aspects
Semilinear Poisson problems in Sobolev-Besov spaces on Lipschitz domains
Full text linkExtending recent work for the linear Poisson problem for the Laplacian in the framework of Sobolev-Besov spaces on Lipschitz domains by Jerison and Kenig [16], Fabes, Mendez and Mitrea [9], and Mitrea and Taylor [30], here we take up the task of developing a similar sharp theory for semilinear problems of the type [Delta]u-N(x, u) = F(x), equipped with Dirichlet and Neumann boundary conditions
Boundary value problems for second order elliptic operators with complex coefficients
Full text linkThe theory of second order complex coeffcient operators of theform L = divA(x)r has recently been developed under the assumption ofp-ellipticity. In particular, if the matrix A is p-elliptic, the solutions u toLu = 0 will satisfy a higher integrability, even though they may not becontinuous in the interior. Moreover, these solutions have the property thatjujp=2-1u 2 W1;2loc . These properties of solutions were used by Dindos-Pipherto solve the Lp Dirichlet problem for p-elliptic operators whose coeffcientssatisfy a further regularity condition, a Carleson measure condition that hasoften appeared in the literature in the study of real, elliptic divergence formoperators. This paper contains two main results. First, we establish solvabilityof the Regularity boundary value problem for this class of operators, in thesame range as that of the Dirichlet problem. The Regularity problem, even inthe real elliptic setting, is more delicate than the Dirichlet problem becauseit requires estimates on derivatives of solutions. Second, the Regularity resultsallow us to extend the previously established range of Lp solvability ofthe Dirichlet problem using a theorem due to Z. Shen for general boundedsublinear operators
Perturbation theory for solutions to second order elliptic operators with complex coefficients and the <em>L<sup>p</sup></em> Dirichlet problem
Full text linkWe establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If L0 = divA0(x)∇+B0(x)·∇ is a p-elliptic operator satisfying the assumptions of Theorem 1.1 then the Lp Dirichlet problem for the operator L0 is solvable in the upper half-space Rn+. In this paper we prove that the Lp solvability is stable under small perturbations of L0. That is if L1 is another divergence form elliptic operator with complex coefficients and the coefficients of the operators L0 and L1 are sufficiently close in the sense of Carleson measures, then the Lp Dirichlet problem for the operator L1 is solvable for the same value of p. As a corollary we obtain a new result on Lp solvability of the Dirichlet problem for operators of the form L = divA(x)∇+ B(x)·∇ where the matrix A satisfies weaker Carleson condition (expressed in term of oscillation of coefficients). In particular the coefficients of A need no longer be differentiable and instead satisfy a Carleson condition that controls the oscillation of the matrix A over Whitney boxes. This result in the real case has been established by Dindos, Petermichl and Pipher
Perturbation and duality of boundary value problems for elliptic and parabolic PDEs
Full text linkIn this thesis we study solvability of the Lp Dirichlet and regularity problems for elliptic and
parabolic PDEs. In the elliptic case we seek to use perturbation theory to expand the class
of operators for which the Lp Dirichlet problem is solvable, to include operators of the form
L = div(A∇) with potentially unbounded antisymmetric part in BMO on bounded chord arc
domains Ω. Specifically, given elliptic operators L0 = div(A0 ∇) and L1 = div(A1 ∇) such that
the Lp Dirichlet problem for L0 is solvable for some 1 < p < ∞; we show that if A0 − A1
satisfies a certain Carleson condition, then the Lq Dirichlet problem for L1 is solvable for some
q ≥ p. Moreover if the Carleson norm is small then we may take q = p.
In the parabolic case we study the relationship between the Dirichlet and regularity problem for
operators of the form L = ∂t − div(A∇) on time cylinders Ω = O × R, where the base O ⊂ Rn
is 1-sided chordarc domain or Lipschitz. Given 1 < p < ∞, we show that if the Lp Dirichlet
problem for the adjoint is solvable (denoted (DL∗ )p′ ) and the Lq regularity problem is solvable
(denoted (RL )q ) for some 1 < q < ∞, then (RL )p holds. This is done in two steps. We first
show the above for q < p, and then interpolate the regularity problem with a suitable endpoint
space to prove that (RL )q implies (RL )r , for 1 < r < q; thus bringing the parabolic setting up
to speed with the elliptic one for Lipschitz cylinders Ω
Jack Alive / Martin Dead : The Location of the "Author" in Jack London\u27s Martin Eden
No full textThis essay is an attempt to read Martin Eden, Jack Londonʼs autobiographical novel, in terms of the inextricable relationship between the author and the protagonist. Critics have often taken the unbalanced plot and the lack of ironic distance between narrator and character in Martin Eden as the technical weakness of London, but this paper argues that the achievement of this novel owes a great deal to the attachment of London to Martin. The unbalanced structure is a necessary product of the severe struggle of the author to kill his romantic alter ego. // Martin, who aspires to win Ruth Morse, tries to cross class boundaries by making a career of a writer. Even after realizing the emptiness of Ruth, who turns out to be nothing but a typical figure of the bourgeoisie, he somehow persists in loving her. The notion underlying here is that, for Martin, love, career and art are fundamentally inseparable. He objects to the aestheteʼs view of Brissenden on account of his separation of art from career. Martinʼs identity and life consist only in the triunity of love/career/art; the alternative is the repudiation of life. Thus, the unnatural delay of his disappointment in love can be regarded as Londonʼs strategy to set the suicide of Martin as the necessary consequence of the story. // By finishing the story and killing Martin, London finally detaches himself from Martin, reconstructs his self, and, unlike Martin, survives as a professional writer. In this sense, Martin Eden is a story about “writerʼs self-reconstruction.
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Robert Martin Tiffin's Mystery Man Newspaper Articles
No full textAdvertiser-Tribune newspaper clippings featuring a story about Robert Martin (written by Nancy Kleinhenz), a local author from Tiffin (Ohio) who wrote under the pseudonym of Lee Roberts, and two of his short stories. Martin wrote mystery novels in his spare time, creating more than 22 mystery novels. For more information about Robert Martin and a list of books go to http://www.mysteryfile.com/RMartin/JBennett.html
Limits Of Transfinite Convergent Sequences Of Derivatives
No full text1 . The paper solves the question whether the limit of transfinite convergent sequence of derivatives is again the derivative. It shows that this problem cannot be solved in the ZermeloFraenkel axiomatic system and that this statement is equivalent to the covering number for Lebesgue null ideal being bigger that @ 1 . In the second part of the paper author proved an analogue of Preiss's theorem [P] for the transfinite sequences of derivatives. 1. Introduction The covergence of transfinite sequences of functions was introduced in the paper [Sie]. Let\Omega be the first uncountable ordinal number, let I be a real interval and f ¸ : I ! R, 1 ¸ !\Omega be a sequence of real functions. We say that f : I ! R is the pointwise limit of this sequence if f ¸ (x) ! f(x) holds for every x 2 I, i.e. 8x 2 T 8" ? 0 9j !\Omega 8¸ j : jf(x) \Gamma f ¸ (x)j ! " We shall denote this convergence by f ¸ ! f or more precisely lim ¸!\Omega f ¸ = f . An important question is whether the pointwise ..
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