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 dedicated in general to the precise mathematical research of difficulties in continuum mechanics, but additionally to difficulties of a basically mathematical nature usually 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".
Read or Download Analysis and Continuum Mechanics: A Collection of Papers Dedicated to J. Serrin on His Sixtieth Birthday PDF
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Extra info for Analysis and Continuum Mechanics: A Collection of Papers Dedicated to J. Serrin on His Sixtieth Birthday
30) 1J(0, x) =x and for all x E E, where tp is a cut-off function so chosen that tp is positive in U. and vanishes identically outside U•. Yl) < <%, a contradiction. This proves our assertion. A few remarks are in order. 9 in [R], pp. 148-153. Also observe that some "compactness" conditions, such as the well-known "Palais-Smale condition", are usually assumed in treating this kind of deformation. In our case, the functional J does not enjoy compactness of that kind in E. However, we get around this difficulty by constructing an explicit path Yl which achieves <% and is naturally compact.
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>, K o· Since g~(tk) = 0, tk ~ T for all k::;:" k o. This proves our assertion. 35), we see that J1(tkUk) < J(Uk) = (Xk + p71 f b(u~ + UJ:+l) + Bk Tp +1• Since (Xk converges to ex, which is smaller than max J 1 (u) uE/k e s:: fJ - fJ, we have for k sufficiently large, where i < (J is some positive constant. Now we can construct a path Yk E Fk based on lk such that max J 1 (u) :S fJ uEl'k e. 14 and is therefore omitted here. By the definition of fJh we then have fJk < fJ - e for all k sufficiently large.
K o· Since g~(tk) = 0, tk ~ T for all k::;:" k o. This proves our assertion. 35), we see that J1(tkUk) < J(Uk) = (Xk + p71 f b(u~ + UJ:+l) + Bk Tp +1• Since (Xk converges to ex, which is smaller than max J 1 (u) uE/k e s:: fJ - fJ, we have for k sufficiently large, where i < (J is some positive constant. Now we can construct a path Yk E Fk based on lk such that max J 1 (u) :S fJ uEl'k e. 14 and is therefore omitted here. By the definition of fJh we then have fJk < fJ - e for all k sufficiently large.