By Professor Dr. Stuart S. Antman, Professor Dr. Haïm Brezis, Professor Dr. Bernard D. Coleman, Professor Dr. Martin Feinberg, Professor Dr. John A. Nohel, Professor Dr. William P. Ziemer (auth.)
The 39 papers during this assortment are committed often to the precise mathematical research of difficulties in continuum mechanics, but additionally to difficulties of a only mathematical nature quite often attached to partial differential equations from continuum physics. all of the papers are devoted to J. Serrin and have been initially released within the "Archive of Rational Mechanics and Analysis".
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Extra resources for Analysis and Continuum Mechanics: A Collection of Papers Dedicated to J. Serrin on His Sixtieth Birthday
In section 3, we establish regularity properties and positivity of the weak solution v, and then adapt to the present problem the powerful method initiated by SERRIN in  and greatly enlarged by GIDAS, NI & NIRENBERG in . A slight, further extension is needed here because C 2 solutions are considered in  and , whereas for HILL'S vortex second derivatives have a finite jump across oA H . The generalized maximum principles in GILBARG & TRUDINGER'S book  enable us to modify the relevant parts of .
4a) tends to zero as IXo I ~ 00; so, therefore, does the norm ofv in Cll-"'(B:J), and this shows that vex) ~ as Ix I -~ 00. (c) Since v is continuous in R S, the set P(v) is open. 1) that ° J cf>(Llv + A) = ° for all cf> E Co(P(v»); P(v) hence -Llv =,1. almost everywhere in P(v). 6) below; in fact, v is real-analytic in P(v). The argument is similar for the set in which vex) < 1- W. (d) This follows from (b) and the definition of P(v). d. 2. 6) in RS. Proof. 1) to be a smooth approximation to the Newtonian kernel in RS.
By the compactness of B R , there is a subsequence rkJ that converges to r;;::; R and as j ~ 00 since Uk converges to u uniformly on BR • Thus, and =$ 0. 6. 2) hold. 1) has a positive radial solution in E,. 8. 7) has a positive radial solution in H6(Rn). 8 must tend to zero exponentially fast at infinity. This standard fact follows from the Radial Lemma and a well known result of T. KATO [K]. § 5.. 8) a natural approach exploits the classical method by maximization (or, equivalently, minimization).
Analysis and Continuum Mechanics: A Collection of Papers Dedicated to J. Serrin on His Sixtieth Birthday by Professor Dr. Stuart S. Antman, Professor Dr. Haïm Brezis, Professor Dr. Bernard D. Coleman, Professor Dr. Martin Feinberg, Professor Dr. John A. Nohel, Professor Dr. William P. Ziemer (auth.)