Partial Differential Equations in Action: From Modelling to Theory (Universitext)

Partial Differential Equations in Action: From Modelling to Theory (Universitext)

Language: English

Pages: 556

ISBN: 8847007518

Format: PDF / Kindle (mobi) / ePub

This book is designed as an advanced undergraduate or a first-year graduate course for students from various disciplines. The main purpose is on the one hand to train students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other hand to give them a solid theoretical background for numerical methods. At the end of each chapter, a number of exercises at different level of complexity is included

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covariance matrix equal to tIn , whose density is |x|2 1 − 2t Γ (x, t) = Γ 21 (x, t) = e . n/2 (2πt) Moreover, (2.106) is independent of any event occurred at any time less than s. For instance, the two events {B (t2 ) − B (t1 ) ∈ A1 } are independent if t0 < t1 < t2 . 38 See Appendix B. {B (t1 ) − B (t0 ) ∈ A2 } 2.6 Multidimensional Random Walk 61 • Transition function. For each Borel set A ⊆ Rn a transition function P (x, t, A) = P x {Bx (t) ∈ A} is defined, representing the probability

Ω. We postpone the proof to the end of the Section 3.4. 3.3.3 Maximum principles As in the discrete case, a function satisfying the mean value property in a domain4 Ω cannot attain its maximum or minimum at an interior point of Ω, unless it is constant. In case Ω is bounded and u (non constant) is continuous up to the boundary of Ω, it follows that u attains both its maximum and minimum only on ∂Ω. This result expresses a maximum principle that we state precisely in the following theorem. 4

of the main boundary value problems and state some basic results. The reader can find complete proofs and the integral formulation of more general or different problems in the literature at the end of the book (e.g. Dautrait-Lions, vol 3, 1985). 142 3 The Laplace Equation 3.7.1 The double and single layer potentials The last integral in (3.56) is of the form D (x;μ) = μ (σ) ∂ν σ Φ (x − σ) dσ (3.87) ∂Ω and it is called the double layer potential of μ. In three dimensions it represents the

properties of D (x;μ), that we state in the following theorem. lim z→x, z∈Ω Theorem 3.17. Let Ω ⊂ Rn be a bounded, C 2 domain and μ a continuous function on ∂Ω. Then, D (x;μ) is harmonic in Rn \∂Ω and the following jump relations hold for every x ∈ ∂Ω : lim z→x, z∈Rn \Ω and lim z→x, z∈Ω 1 D (z;μ) = D (x;μ) + μ (x) 2 1 D (z;μ) = D (x;μ) − μ (x) . 2 (3.91) (3.92) Proof (Sketch). If x ∈ / ∂Ω there is no problem in differentiating under the integral sign and, for σ fixed on ∂Ω, the function

line x= t . 2 Hence, the function w (x, t) = 0 1 x< x> t 2 t 2 is another weak solution (Fig. 4.19). As we shall see, this shock wave has to be considered not physically acceptable. The example shows that the answer to question Q2 is negative and question Q3 becomes relevant. We need a criterion to establish which one is the physically correct solution. 4.4 Integral (or Weak) Solutions 183 Fig. 4.19. A non physical shock 4.4.4 The entropy condition From Proposition 4.2 we have seen

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