Partial Differential Equations (Graduate Texts in Mathematics)
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This book offers an ideal introduction to the theory of partial differential equations. It focuses on elliptic equations and systematically develops the relevant existence schemes, always with a view towards nonlinear problems. It also develops the main methods for obtaining estimates for solutions of elliptic equations: Sobolev space theory, weak and strong solutions, Schauder estimates, and Moser iteration. It also explores connections between elliptic, parabolic, and hyperbolic equations as well as the connection with Brownian motion and semigroups. This second edition features a new chapter on reaction-diffusion equations and systems.
have the following important consequence of Theorem 6.1.1, a global existence theorem: Corollary 6.1.1. Under the assumptions of Theorem 6.1.1 , suppose that the solution of (6.1.8) satisfies the a priori bound (6.1.23) for all times t for which it exists, with some fixed constant K. Then the solution u(x,t) exists for all times . Proof. Suppose the solution exists for . Then we apply Theorem 6.1.1 at time T instead of 0, with initial values u(x, T) in place of the original initial
the test functions η, we can actually require that h − η ≤ ( ≥ )0 and . Derivatives then are only evaluated for test functions that touch h at the point in question, but not for h itself. First of all, this solution concept is consistent in the sense that when a viscosity h is smooth, it is a classical solution of (9.2.5). This is trivial; as in that case, we may use the test function η = h so that has both a local maximum and minimum at any point, and the two inequalities in (9.2.12) then
to , convex and lower semicontinuous. Then, for every , and , (10.6.4) is realized by a unique , i.e., (10.6.5) and if remains bounded as , then exists and minimizes I, i.e., Proof. We first verify the auxiliary statement about the uniqueness and existence of . We let be a minimizing sequence for (10.6.4), i.e., For , we put We then have (10.6.6) by the convexity of I and the general Hilbert space identity (10.6.7) for any , which is easily derived from expressing the norm
following result: Lemma 11.2.2. Let , and suppose there exists K < ∞ with and (11.2.2) for all h > 0 and with . Then the weak derivative D i u exists and satisfies (11.2.3) Proof. For and ( is the closure of ), we have as h → 0. Thus, we also have Since is dense in L 2(Ω), we may thus extend to a bounded linear functional on L 2(Ω). According to the Riesz representation theorem as quoted in the appendix, there then exists v ∈ L 2(Ω) with Since this is precisely the equation
rules for manipulating difference quotients, like the product rule (11.3.25) For example, (11.3.26) As before, we use as a test function in place of v, and in the case , , we obtain (11.3.27) With (11.3.25) and Lemma 11.2.1, this yields (11.3.28) i.e., an analogue of (11.2.17). Since because of the ellipticity condition (A11.3), we have the estimate we can then proceed as in the proofs of Theorems 11.2.1 and 11.2.2. Readers so inclined should face no difficulties in supplying the