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By V.S. Vlasov

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By (1 ), u:5 v ; so in O (3 ) for any e > O. It follows that u cannot exceed M in 0 ; if it did, then, for sufficiently small E, (3) would be violated. T he above argument provides an alternate proof that a harmonic function on a bounded region 0 must attain its maximum on the boundary of O. 7 Show that if u(x, y ) is harmonic in a bounded region nand u is continuously differentiable in fl, then /Vu /2 attains its maximum on S, the boundary of n. Let w == JVu j2 = u; + u ~ . Since u is C 1 in 0, w is continuous on S.

20). 2 MAXIMUM-MINIMUM PRINCIPLES (PARABOLIC PDEs) Neither the wave equation nor hyperbolic equations in general satisfy a maximum-mInimum principle, but the heat equation and parabolic equations of more general form do so. Let n denote a bounded region in R3 whose boundary is a smooth closed surface S. Suppose u(x, y, z, t) to be continuous for (x, y, z) in nand O:s t:s T; for short, in n X [0, T]. Let Ms == max {u(x, y, z, t): (x, y, z) on Sand O:s t:s T} Mo == max {u(x, y, z, t): (x, y, z) in nand t = O} M == max {Ms, Mo} and let m s' m o' and m denote the corresponding minimum values for u.

A) Find plane wave solutions for - co < x, y < co, t >0 (b) Are there any values of p for which u(x, y, t) = sin (x/ a ,) cos (y/ a2) sin pt is a (standing wave) solution of the above equation? 20 Consider the problem = Utr 0< x < 1, 0 < t < T U xx u(x, 0) = u(x, T) = 0 O

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BASIC DIFFERENTIAL EQUATIONS IN GENERAL THEORY OF ELASTIC SHELLS by V.S. Vlasov


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