By Shuxing Chen

ISBN-10: 9814304832

ISBN-13: 9789814304832

The ebook offers a finished assessment at the concept on research of singularities for partial differential equations (PDEs). It features a summarization of the formation, improvement and major effects in this subject. the various author's discoveries and unique contributions also are integrated, equivalent to the propagation of singularities of strategies to nonlinear equations, singularity index and formation of shocks

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**Sample text**

41), assume that in a neighborhood ω of (x0 , ξ0 ) the symbol a(x, ξ ) of the operator A has N0 simple real eigenvalues λ1 , · · · , λN0 , m+ complex eigenvalues with positive imaginary part, m− complex eigenvalues with negative imaginary part. 40) into the almost diagonal form. Assume that on xn = xn0 (0 < xn0 < ) the equality W F (U )∩γj = ∅ holds for j = j1 , · · · , jN1 . Moreover, assume that (x0 , ξ0 ) ∈ W F (h), BS −1 is elliptic with respect to Wi1 , · · · , WiN0 −N1 , W + is elliptic, where {i1 , · · · , iN0 −N1 } = {1, · · · , N } \ {j1 , · · · , jN1 }, then W F (U ) ∩ γj = ∅ holds for any j ≤ N0 .

Based on the above preparations we can describe the singularity of u on the manifold with boundary by using boundary wave front set W Fb (u). For any point, which is not on ∂T ∗ M , W Fb (u) is nothing but W F (u). For the point on T ∗ ∂M , W Fb (u) is the complimentary set of the points, which are microlocally regular up to the boundary. Here a point (y0 , η0 ) ∈ T ∗ ∂M is “microlocally regular up to the boundary” means that there is a pseudodifferential operator ψ(y, Dy ) defined in a neighborhood of (y0 , η0 ), such that for some > 0, ψ(y, Dy )u(x, y) is a C ∞ ([0, ] × Rn ) function.

On Σ2b j≥k point with Hp22 > 0 is called diffractive point and is denoted points with Hp22 > 0 is called gliding point, and is denoted (3) points in Σb is called glancing points of higher order. by Σ2− b , by Σ2+ b . Next we indicate that at any glancing point the bicharacteristic strip is tangential to the boundary. Indeed, assume that the operator P has been reduced to the form with the principal symbol ξ 2 + r(x, y, η), then Hp x = 2ξ, Hp2 x = −2rx . Hence rx < 0 at diffractive point. The bicharactristic strip of P is defined by dx dy dξ dη = 2ξ, = rη , = −rx , = −ry .

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