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Contents:

1 limited minimization
1. 1 Preliminaries. .. ..
1. 2 restricted minimization
1. three twin strategy . . . . . . .
1. four Minimizers with the least power .
1. five software of twin approach . ,.
1. 6 a number of recommendations of nonhomogeneous equation.
1. 7 units of constraints . . . . . . . .
1. eight restricted minimization for Ff .
1. nine Subcritical challenge . .. .. .
1. 10 software to the p-Laplacian .
1. eleven severe challenge . . .
1. 12 Bibliographical notes. . . . .

2 purposes of Lusternik-Schnirelman thought
2. 1 Palais-Smale situation, case p '# q
2. 2 Duality mapping . . . . . . . . . .
2. three Palais-Smale situation, case p = q
2. four The Lustemik-Schnirelman idea .
2. five Case p > q
2. 6 Case. p < q . .. .. .. .. .. . 2. 7 Case p = q . .. .. .. .. .. . 2. eight The p-Laplacian in bounded area 2. nine Iterative development of eigenvectors 2. 10 serious issues of upper order 2. eleven Bibliographical notes. . . . . . . . . 3 Nonhomogeneous potentials 3. 1 Preliminaries and assumptions 3. 2 restricted minimization . . 3. three program - compact case. 3. four Perturbation theorems - noncompact case 3. five Perturbation of the useful a - noncompact case. 3. 6 life of infinitely many ideas . . . . . . . . 3. 7 basic minimization - case p > q .
3. eight Set of constraints V . .. .. .. .
3. nine program to a serious case p = n
3. 10 Technical lemmas . . . . . . . . .
3. eleven life consequence for challenge (3. 34)
3. 12 Bibliographical notes. . . . . . .

4 Potentials with covariance
4. 1 Preliminaries and limited minimization
4. 2 twin procedure . . . . . . . . . . . . .
4. three Minimization topic to constraint V . . . .
4. four Sobolev inequality . . . . . . . . . . . . .
4. five Mountain go theorem and restricted minimization
4. 6 Minimization challenge for a procedure of equations .
4. 7 Bibliographical notes. . . . . . . . . . . . . . .

5 Eigenvalues and point units
5. 1 point units . .. .. .. .. .. ..
5. 2 Continuity and monotonicity of a .
5. three The differentiability houses of a
5. four Schechter's model of the mountain cross theorem
5. five common for solvability of (5. eleven)
5. 6 homes of the functionality K(t) .
5. 7 Hilbert house case . . . . . . .
5. eight program to elliptic equations
5. nine Bibliographical notes. . . . . .

6 Generalizations of the mountain go theorem
6. 1 model of a deformation lemma . . . . . .
6. 2 Mountain cross replacement . . . . . . . . .
6. three results of mountain go replacement
6. four Hampwile substitute. . . . . . . . . . . .
6. five Applicability of the mountain go theorem
6. 6 Mountain cross and Hampwile substitute
6. 7 Bibliographical notes. . . . . . . . . . .

7 Nondifferentiable functionals 167
7. 1 proposal of a generalized gradient . . . . . . . . . . . . 167
7. 2 Generalized gradients in functionality areas. . . . . . . . . 172
7. three Mountain cross theorem for in the neighborhood Lipschitz functionals . 174
7. four results of Theorem 7. three. 1 . . . . . . . . . . . . . 181
7. five software to boundary price challenge with discontinuous nonlinearity 183
7. 6 decrease semicontinuous perturbation . . . . . . . . . . . . . . 185
7. 7 Deformation lemma for functionals fulfilling situation (L) . . . . . . 188
7. eight program to variational inequalities
7. nine Bibliographical notes. . . . . . . . .

8 focus compactness precept - subcritical case 198
8. 1 Concentration-compactness precept at infinity - subcritical case 198
8. 2 limited minimization - subcritical case . . . . . . . . 2 hundred
8. three limited minimization with b ¥= const, subcritical case . 205
8. four Behaviour of the Palais-Smale sequences . 211
8. five the outside Dirichlet challenge . . . . . . 215
8. 6 The Palais-Smale situation . . . . . . . 218
8. 7 Concentration-compactness precept I . 221
8. eight Bibliographical notes. . . . . . . . . . . 223

9 focus compactness precept - serious case 224
9. 1 severe Sobolev exponent . . . . . . . . 224
9. 2 Concentration-compactness precept II . . 228
9. three lack of mass at infinity. . . . . . . . . . . 229
9. four restricted minimization - serious case . 233
9. five Palais-Smale sequences in severe case . . 237
9. 6 Symmetric options . . . . . . . . . . . . . . . . . . 244
9. 7 feedback on compact embeddings into L 2* (Q) and L okay (}Rn) . 250
9. eight Bibliographical notes. . . . . . . . . . . . . . . . . . . . . . 252

Appendix
A. l Sobolev areas . . . . . . . . . . . . . . . . . . . . . .
A. 2 Embedding theorems . . . . . . . . . . . . . . . . . . .
A. three Compact embeddings of areas wI,p(}Rn) and DI,p(}Rn)
A. four stipulations of focus and uniform decay at infinity
A. five Compact embedding for H,1 (}Rn) .
A. 6 Schwarz symmetrization
A. 7 Pointwise convergence.
A. eight Gateaux derivatives

Bibliography

Glossary

Index

Additional info for Applications of potential theory in mechanics. Selection of new results

Example text

8) 46 CHAPTER 1, Description of the new method ρ ∞ ρ0dρ0 x 1  χ( a ,ρ,φ) d  x 2 ∆ v (ρ ,φ). ⌠ ⌠ σ(ρ,φ) = − 2 2 L + 2 2 1/2 2 2 1/2 0 π (ρ − a 2)1/2 ⌡ (ρ − x ) ⌡ (ρ0 − x ) ρρ0   a x It should be noticed that the first term in while the second term vanishes at the edge a harmonic function, the second term in represented by the first term only. 9) becomes singular when ρ→ a , of the disc. 8). The 1  χ( a ,ρ,φ)  π2 (ρ2 − a 2)1/2  2π ∞ ∆ v (ρ0,φ0) ρ0dρ0dφ0 (ρ2 − a 2)1/2(ρ20 − a 2)1/2  1⌠⌠ -1 + tan .

5). 5) yields, after changing the order of 64 CHAPTER 1, Description of the new method integration: d l 20( x ) ρ0 r rdr z  ⌠ ⌠ ⌠ I1 = dψ , ψ−φ0 3 . 8), the following result can be obtained, after integration with respect to ψ: a I 1 = 4⌠ x d l 20( x ) dr ⌠ 2 2 1/2 2 1/2 ⌡ [ l 20( x ) − ρ0] ⌡ ( x − r ) 0 0 r d tdt ⌠ 2 d r ⌡ ( r − t 2)1/2 0 l 1( t ) t ρ0 [ t 2 − l 21( t )]1/2  × λ 2 , φ−φ0. 6 Some fundamental integrals [ x 2 − l 21( x )]1/2 [ x 2 − l 210( x )]1/2 a l ( x) l ( x)  1 10 λ , φ−φ d x .

16) is l 2vn 1 x 2nd x Γ( n + 1) V (ρ,φ, z ) = cosn φ ⌠ 2 2 1/2 √π ρn Γ( n + 1) ⌡ (ρ − x ) 0 2 2 2 1/2 l2 − a2 2Γ( n + 1) ( l 2 − a ) 1 1 3 2   . 34) can be expressed in elementary functions (Bateman and Erdelyi, 1955) F( 1 1 3 (1 − ζ)n+1/2 dn  ζn-1/2 sin-1√ζ. − n , ; ; ζ) = 2 2 2 Γ( n + 1) dζn √1 − ζ  Example 2. Let the charge distribution be prescribed σ(ρ,φ)=σnρncosn φ, σn=const. 4 Internal mixed boundary value problem for a half-space 2 n + 3 dn 1 1 1 + √ζ 5 3 . 37) Bibliographical note.

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Applications of potential theory in mechanics. Selection of new results by V.I. Fabrikani


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