Applied Analysis I - ( Advanced PDE I ) ( Math 940 , Fall 2014

function, 51 Riemann integral, 51 adjoint bilinear form, 65 almost separably valued, 53 Ascoli Theorem, 8 Banach space reflexive, 6 separable, 2 Banach-Steinhaus Theorem, 5 Bessel’s inequality, 9 biharmonic, 61 bilinear form, 55, 58 bounded, 58 strongly positive, 58 Bochner-Pettis Theorem Theorem, 53 Bounded Inverse Theorem, 6 Calculus in W 1,p(I;X) Theorem, 54 Campanato ’63 Theorem, 86 Campanato Lemma Theorem, 92 Caratheodory property, 13 Cartesian product, 3 Cauchy sequence, 2 Cauchy-Schwarz inequality, 4 classical solution, 140 Closed Graph Theorem Theorem, 127 co-area formula, 120 coercivity, 59 compact linear operator, 65 compactly embedded, 45 Construction of approximate solutions Theorem, 99, 112 continuation method, 11 continuous, 51 Contraction Mapping Theorem, 11 convex, 18 critical point, 18, 20 Difference quotient and weak derivatives Theorem, 48 differentiable, 51 differentiable on I, 51 differential operator uniformly elliptic, 55 divergence form, 55 dual space, 6 second dual space, 6 Duhamel formula, 141 Ehrling’s inequality, 21 eigenfunctions, 10, 67 eigenspace, 68 eigenvalue, 67 Eigenvalue Theorem Theorem, 67 eigenvalues, 10 eigenvectors, 10 Energy estimates Theorem, 122 equicontinuous, 8 Estimates for W 1,p, n < p ≤ ∞ Theorem, 44 evolution equations, 97 Existence of approximate solutions Theorem, 122 Existence of solution by Fourier transform Theorem, 124 Existence of weak solution by vanishing viscosity Theorem, 124 Existence Theorem, 114 extreme, 19 maximal, 20 minimal, 20 finite propagation speed, 119 Finite propagation speed Theorem, 119 First Existence Theorem for weak solutions Theorem, 59 fixed point, 11 formal adjoint, 65 Fourier series, 9 Fourier transform, 49 Fourier transforms, 62 Frechet differentiable, 13