Corrigendum on 'Elliptic-Parabolic Equations of the Second Order'

Phillips, R.; Sarason, Leonard

doi:10.1512/iumj.1969.18.18019

Cited by 20 publications

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“…If <j>k is not nonnegative, we set and another application of (1.13) together with Schwarz' inequality and the estimate \\u\\r. < c||A|| yields the a priori estimate ||VM||2a +(K-1)||«||B +||«||2r +||M||2r < Cx(\\jf_m) +||g||r + \\hfr\ (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) where || || _, is the norm in H ' '(£2), the space dual to H '(ß) with norm llMlli = (llMf + !IVmII2)' 2> ll"ll-i = sup ("'ü)-Wli-l Note that if <¡>k > 0, k = 1, . .…”

Section: Withmentioning

confidence: 99%

“…Since T" = <¡>, cJk + ckj < 0 uniformly along TJk. Along TJk, we have Jk/Jj=\ckj/cjk\, (4)(5)(6) where Jj is the restriction to Tj of /IC,!/^«^. Hence Je decreases as we follow an integral curve 5 of the field Ce in P\ U H-Therefore we can choose re to be piecewise smooth, identically 1 in U H, r > r0 > 0, say, and such that r'(Je ° <¡>~l) decreases monotically along í.…”

Section: Withmentioning

confidence: 99%

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Second-order equations of fixed type in regions with corners. I

Sarason¹

1980

Trans. Amer. Math. Soc.

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Abstract.A class of well-posed boundary value problems for second order equations in regions with corners and edges is studied. The boundary condition may involve oblique derivatives, and edge values may enter the graph of the associated Hilbert space operator. Uniqueness of weak solutions and e?cistence of strong solutions is shown.Introduction. In this paper, we study boundary value problems for second order linear partial differential equations of standard type (elliptic, parabolic, or hyperbolic), in regions with corners and edges. Our results also cover first order two-by-two systems. Only interior corners with angles normalized as 77/2 have been considered, but much of the theory would appear to extend easily and with minimal changes to problems with exterior corners. This work originated as an attempt to understand formulas developed in [5] in the two-dimensional constant coefficient elliptic case.In Chapter I, we develop the existence and uniqueness theory for elliptic problems with variable coefficients. The principal parts of the differential operator and of the boundary conditions are real. The boundary conditions for the second order equation are coercive and dissipative, in the sense that they lead to quadratic a priori estimates. We begin with the basic (Gárding) inequality (Theorem 1.1), derived by integrations by parts combined with an estimate of the boundary values of the solution u in terms of the H ' norm of u in the interior Q of the region under consideration. We are indebted to Stanley Osher for pointing out that Gârding's inequality is available even with corners. If the boundary conditions involve oblique derivatives, then an additional integration by parts of the boundary term is required, leading in general to a quadratic term in the edge values of u. The signature of this term determines (locally) whether or not u is to be prescribed along the edge (this effect does not occur in dissipative problems for first order systems). In §2, we use duality to show the existence of weak solutions (Theorem 2.1). In §3, we consider problems admitting the maximum principle. The existence of solutions satisfying the maximum principle when the problem is not Dirichlet,

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Section: Withmentioning

confidence: 99%

Section: Withmentioning

confidence: 99%

Second-order equations of fixed type in regions with corners. I

Sarason¹

1980

Trans. Amer. Math. Soc.

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“…Recall [2], [9] that if (1.4) holds and the a^x) have continuous second derivatives then there exists a matrix σ(x) satisfying (B). Consider the stochastic differential system (see [5], [6] for the relevant theory)…”

Section: (C) σ \σ Ij (X)\ + ±\B I (X)\^c(l + \X\)mentioning

confidence: 99%

Uniqueness for the Cauchy problem for degenerate parabolic equations

Friedman¹

1973

Pacific J. Math.

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“…Kannai [10] has given a virtually complete characterization of hypoelliptic ordinary differential operators. LEMMA 1.1 [13]. o~ij(x) are Lipschitz continuous in Rd.…”

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confidence: 99%

Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form

Amano¹

1984

Trans. Amer. Math. Soc.

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ABSTRACT. In the theory of partial differential equations, there is no explicit representation of solutions for general degenerate elliptic-parabolic equations. However, Stroock and Varadhan [15] have obtained a stochastic representation for such a wider class of equations in L°° space. In this paper we establish, by using Stroock and Varadhan's stochastic representation, a method which enables us to construct solutions with singularities of second order equations with semidefinite characteristic form. Our theorems are not probabilistic paraphrases of the results obtained in the theory of partial differential equations. In fact, each assumption of the theorems is much weaker than any assumption of corresponding known results. Introduction.As is well known, it is one of many interesting problems to characterize the (analytic) singularities of solutions of second order equations with semidefinite characteristic form, and this problem was investigated by many authors. However, most of them gave only sufficient conditions for (analytic-) hypoellipticity. In this paper we shall give sufficient conditions under which a certain class of second order operators with semidefinite characteristic form are not (analytic-) hypoelliptic. The proof depends on Stroock-Varadhan's stochastic representation of solutions of degenerate elliptic-parabolic equations; we solve the Dirichlet problem with singular boundary condition by using probabilistic methods and obtain a solution with (analytic) singularities.Let G be an open set in Rd and let us consider a differential operator

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Corrigendum on 'Elliptic-Parabolic Equations of the Second Order'

Cited by 20 publications

References 0 publications

Second-order equations of fixed type in regions with corners. I

Second-order equations of fixed type in regions with corners. I

Uniqueness for the Cauchy problem for degenerate parabolic equations

Stochastic representation and singularities of solutions of second order equations with semidefinite characteristic form

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