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 regularly to the precise mathematical research of difficulties in continuum mechanics, but in addition to difficulties of a in basic terms mathematical nature in general hooked up 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**

**Example text**

Set = leu) = ! IIul1 2 f H(x, u) - R" where H(x, u) = u f o hex, t) dt. We choose e > 0 in El (defined in § 2) so that l(te) < 0 and l(te) < 0 for all t > I. Observe that this change does not alter the value IX. ) Let f3 = inf max l(u) , yEr uEy M, = {UE E\ {O} IIIul1 2 = R£ uf(x, u)} Mk = {UE E\ {O} IIIul1 2 = R£ uh(x, U)}. 1) we see that hex, u)/u is strictly increasing in u > 0 if hex, u) > 0; in fact, hex, u)/u' +- is nondecreasing. -Y. -M. NI hex, u(x» > O} is not empty, then the half-line lu = {tu I t ~ O} intersects Mh at exactly one point tu.

Take K(x) Ixl > R, if Ixl < R; Q(X) if = {0 the result follows from the preceding corollary. d. 15. 17. 18) sup JQu~+ lIulI=1 Rn 1 > lim [sup R-+oo J Qu~+l] . 18), but lim Q(x) = 0 is obviously included. Also, if Q(x) > ao > 0 for Ix I < Ro Ixl ..... 8) can be solved. 19. 18) holds and equation If lim Q(x) = inf Q(x), Jxl ..... 8) has a positive solution in H~(Rn). -Y. -M. NI Proof. Set m If Q lim Q(x) = Ixl-+OO * constant, then it is clear that ,x - JQU~+I > sup = inf Q(x). xERn Xm 'c-, sup Jmu~+1 Ilull = 1 Rn Ilull = I Rn since Xm is attained by some positive function.

18), but lim Q(x) = 0 is obviously included. Also, if Q(x) > ao > 0 for Ix I < Ro Ixl ..... 8) can be solved. 19. 18) holds and equation If lim Q(x) = inf Q(x), Jxl ..... 8) has a positive solution in H~(Rn). -Y. -M. NI Proof. Set m If Q lim Q(x) = Ixl-+OO * constant, then it is clear that ,x - JQU~+I > sup = inf Q(x). xERn Xm 'c-, sup Jmu~+1 Ilull = 1 Rn Ilull = I Rn since Xm is attained by some positive function. On the other hand, given any e > 0, there is an r large enough that sup x, - J Qu~+l ~ (m + e) Iluil=llxl>' ,sup J U~+l l,ull=llxl>,