ClearAll

(10.1)  Differential Operators in polar coordinates

dx[u_] := Cos[t]*D[u, r] - Sin[t]*D[u, t]/r
dy[u_] := Sin[t]*D[u, r] + Cos[t]*D[u, t]/r
dxx[u_] := dx[dx[u]]
dyy[u_] := dy[dy[u]]
dxy[u_] := dx[dy[u]]
dyx[u_] := dy[dx[u]]
lap[u_] := dxx[u] + dyy[u]
hes2[u_] := dxx[u]^2 + dyy[u]^2 + 2*dxy[u]^2

(10.2) Candidate Modes  
Notebook parameters   (a,b,α,β,γ,δ,λ)   vs  file.TEX parameters (A,B,c1,c2,c3,c4)
a = 1/3 A , b = 1/3 B,
   a 𝔄 = c4 ,  a 𝔊 = c2,   b 𝔅 = c3,  b 𝔇 = c1, 
   (*  u is a generic mute variable , used in several different contexts 
    parameter A in the paper is 3 times parameter  a  in the Notebook 
    parameter B in the paper is 3 times parameter  b  in the Notebook   *)

us := r^(3/2)*Sin[t/2]
vs := r^(3/2)*Sin[3*t/2]
uc := r^(3/2)*Cos[t/2]
vc := r^(3/2)*Cos[3*t/2]
u :=  
      a*(𝔄*us + 𝔊*vs) + 
        b*(𝔅*uc + 𝔇*
                vc)


      
ω :=  ( 
        Sin[t/2]  -  
          5/3 * Sin[
              3*t/2])


      
w :=  ( Cos[t/2]  -  7/3  * Cos[3*t/2])

u

(10.3) Euler conditions: formula (4.3) of Theorem  4.3  at  θ  = π

dy[dy[u]] /. t -> π

Simplify[%]

(10.4) Euler condions:  formula (4.3) of Theorem 4.3   at  θ  = -π

dy[dy[u]] /. t -> -π

Simplify[%]

(10.5) Euler conditions: formula (4.4) of Theorem 4.3   at  θ  = π

dy[dy[dy[u]]] + 2*dx[dx[dy[u]]] /. t -> π

Simplify[%]

(10.6) Euler conditions: formula (4.4) of Theorem 4.3   at  θ  = -π

dy[dy[dy[u]]] + 2*dx[dx[dy[u]]] /. t -> -π

Simplify[%]

(10.7)  Choice of prmeters 𝔄, 𝔅, 𝔊, 𝔇  to be compiled in order to Plot Graphics

𝔄 = 3; 𝔅 = 
      3; 𝔊 = -5; 𝔇 = -7;

u

Evaluation of candidate (normalized and weighted by Cos(θ) ) energy in a disk (in order to force equipartition of energy)
                 and
Evaluation of extremality conditions  at crack-tip  (in order to fulfil conditions of Theorem 5.5)

(10.8)  Hessian of u ( normaliztion of candidte W)    hes2[u] = |∇^2 u |^2

Simplify[r*hes2[u]]

(10.9)  Squared hessian ( |∇^2 u |^2 )  of u integrated over the disk B_R   
(here we compute 9 times second equation of (7.36) in Section 7)

C1 = Simplify[ Integrate[%, {r, 0, R}, {t, -Pi, Pi}] ]

(10.10)  k_1+k_2  of u 
(here we compute 9 times equation of (7.32) in Section 7)

\(Integrate[
        R*Simplify[
            hes2[u]*Cos[t] + 
              2*	dx[u]*\((\ 
                  D[dxx[u]*Cos[t]*dx[t] + dyy[u]*Sin[t]*dy[t] + \ 
                        dxy[u]*\((Cos[t]*dy[t] + Sin[t]*dx[t])\), t] + 
                    D[lap[u], r])\)], {t, \(-Pi\), Pi}] /. r -> R\)

C2 = Simplify[%]

(10.11)  Solution of algebric system (equipartition of energy and k_1+k_2=α )

sol = 
      Solve[{C1 == α*R, C2 == α}, {a, 
          b}]

Computations related to higher order modes       uj_h ,  h > 0

Definiton of an integer variable h

h ∈ Integers

(10.12) Euler conditions for uj_h (Theorem 4.3  formula (4.3) )   at  θ  = π

\(\(\(
\)\(us = r^\((h + 3/2)\)*Sin[\((h - 1/2)\)*t]

    vs = r^\((h + 3/2)\)*Sin[\((h + 3/2)\)*t]

    uc = r^\((h + 3/2)\)*Cos[\((h - 1/2)\)*t]

    vc = r^\((h + 3/2)\)*Cos[\((h + 3/2)\)*t]

    u\ \  = \ 
      a*\((𝔄*us + 𝔊*vs)\)\  + \ 
        b*\((𝔅*uc + 𝔇*vc)\)\)\)\)

dy[dy[u]] /. t -> π

Simplify[%]

(10.13) Euler conditions for uj_h  ( Theorem 4.3 formula (4.3) )   at  θ  = - π

dy[dy[u]] /. t -> -π

Simplify[%]

(10.14) Euler conditions for uj_h  ( Theorem 4.3 formula (4.4) )  at  θ  =  π

dy[dy[dy[u]]] + 2*dx[dx[dy[u]]] /. t -> π

Simplify[%]

(10.15) Euler conditions for uj_h  ( Theorem 4.3 formula (4.4) )   at  θ  = - π

dy[dy[dy[u]]] + 2*dx[dx[dy[u]]] /. t -> -π

Simplify[%]

Integration on the boundary of a small disk : 
Theorem 5.5 - Crack Tip (with  μ=0 , e.g. without g)

u

(10.16) Evalution of  k_1  (NEL FILE TEX  k_1 e k_2 sono definiti solo per h=0)

Simplify[
        Integrate[
          R*Simplify[hes2[u]*Cos[t]], {t, -Pi, 
            Pi}] ] /. r->R

(10.17) Evaluation of  k_2 (NEL FILE TEX  k_1 e k_2 sono definiti solo per h=0)

\(Simplify[\ 
        Integrate[
          R*Simplify[
              2*	dx[u]*\((\ 
                  D[dxx[u]*Cos[t]*dx[t] + dyy[u]*Sin[t]*dy[t] + \ 
                        dxy[u]*\((Cos[t]*dy[t] + Sin[t]*dx[t])\), t] + 
                    D[lap[u], r])\)], {t, \(-Pi\), Pi}]\ ]\  /. 
      r -> R\)

(10.18) Evalution of  k_1+k_2

\(Integrate[
        R*Simplify[
            hes2[u]*Cos[t] + 
              2*	dx[u]*\((\ 
                  D[dxx[u]*Cos[t]*dx[t] + dyy[u]*Sin[t]*dy[t] + \ 
                        dxy[u]*\((Cos[t]*dy[t] + Sin[t]*dx[t])\), t] + 
                    D[lap[u], r])\)], {t, \(-Pi\), Pi}] /. r -> R\)

C2 = Simplify[%]

(10.19)  Deduction of parameters 𝔄, 𝔅, 𝔊, 𝔇 :
Solution of algebric system (equipartition of energy and k_1+k_2=α )

sol = 
      Solve[{C1 == α*R, C2 == α}, {a, 
          b}]

coefficients  a=a_h  and b=b_h  for  uj_h  
(the above system has four solutions:  ( ± a,  ± b ) we chose the fourth one (+a,+b) related to the candidate W )
A_h = 3 a_h , B_h = 3 b_h

Simplify[sol[[4]]]

coefficients  a  and b  for   u0_h

Evaluate[%]   /.  h -> 0

{a -> (Sqrt[(7/(579 π))]) (Sqrt[α]), 
      b -> ((Sqrt[α])/(3 (Sqrt[193 π])))}

u = u /. %

Simplify[u]

(10.20) Evaluation of laplacian of the candidates   W  and Φ  (combination of modes at index  h=0) :   (Subscript[Δ, x]) W ,   (Subscript[Δ, x]) Φ

lap[u]

Evaluation of laplacian of the candidate  W  (h=1)   ( u= W with α=1)

Simplify[lap[u]]

W[r_, 
        t_]  :=   r^(3/2) *  (  1  / ( 
              Sqrt[193 * 
                  Pi]   )  ) * 
                                          (   Sqrt[21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )   +  
                                                                   (  Cos[t/2]    -  
                7/3 * Cos[
                    3 * t/
                        2] )       )

lap[W[r, t]]

Simplify[%]

Evaluation of laplacian of the candidate   Φ    (with α=1)

Φ[r_, 
        t_]  :=   r^(3/2) *  (  1  / ( 
              Sqrt[193 * 
                  Pi]   )  ) * 
                                          (   Sqrt[21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )   -  
                                                                   (  Cos[t/2]    -  
                7/3 * Cos[
                    3 * t/
                        2] )       )

lap[ Φ[r, t]]

Simplify[%]

THE CANDIDATE IS BIHARMONIC

(10.21) Evaluation of  bi-laplacian of the candidte W  ( (Δ^2) W  )

lap[lap[u]]

Simplify[lap[lap[u]]]

(10.22)  Candidate  W in polar coordinates    (sign(A)=sign(B))

W = 
      Sqrt[α/(193*Pi)]*
        r^(3/2)*(Cos[t/2] - (7/3)*Cos[3*t/2] + 
            Sqrt[21]*(Sin[t/2] - (5/3)*Sin[3*t/2]))

(10.23)  Re-normalization   u   of  leading coefficients  in W (sum of the two modes)

u = W  /.  α -> 1

(10.24)  Re-normalization of α   in Φ  (difference of the two modes)

(10.25) Graphs of candidates W and Φ  in  a disk (polar coordinates): Figg. 4 e 7

(*     Graph of W in polar coordinates (true coefficients,  normalized α = 1)   *)

ClearAll

W[r_, 
        t_]  :=   r^(3/2) *    1  / ( 
            Sqrt[193 * 
                Pi])  * 
                                          (   Sqrt[
                21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )  +  
                                                                   (  Cos[t/2])    -  
            7/3 * Cos[
                3 * t/
                    2]        )   

    

    ParametricPlot3D[{r * Cos[t] ,  r * Sin[t] ,  W[r, t],  
        Hue[W[r, 
            t]]  },  
              {r, 0, 
        1.1}, {t, -Pi, Pi},   PlotPoints -> 50,  
      ViewPoint -> {-2.651,  -2.271,  1.2}]

(*     Graph of Φ in polar coordinates (true coefficients,  normalized α = 1)   *)

ClearAll

Φ[r_, 
        t_]  :=   r^(3/2) *  1  / ( 
            Sqrt[193 * 
                Pi])  * 
                                          (   Sqrt[
                21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )   -  
                                                                   (  Cos[t/2]    -  
                7/3 * Cos[
                    3 * t/
                        2] )       )   

    
ParametricPlot3D[{r * Cos[t] ,  
        r * Sin[t] ,  Φ[r, t],  
        Hue[Φ[r, 
            t]]  },  
              {r, 0, 
        1.1}, {t, -Pi, Pi},   PlotPoints -> 50,  
      ViewPoint -> {-2.651,  -2.271,  1.2}]

(10.26) Candidate   W   in cartesian coordinates 

candW[x_, 
          y_] = 
1/
            Sqrt[193*2*
                Pi]  (Sqrt[
                  21] *
(Sign[
                        y]*(x^2 + y^2)^(1/2)*
                      Sqrt[(x^2 + y^2)^(1/2) - x] - 
                    5/3*(y*Sqrt[(x^2 + y^2)^(1/2) + x] + 
                          x*Sign[y]*
                            Sqrt[(x^2 + y^2)^(1/2) - 
                                x])) + (x^2 + y^2)*
                Sqrt[(x^2 + y^2)^(1/2) + x] - 
              7/3 (x*Sqrt[(x^2 + y^2)^(1/2) + x] - 
                    Abs[y]*Sqrt[(x^2 + y^2)^(1/2) - 
                          x]))

      
\(\(1\/\@\(386\ π\)\) \((\((x\^2 + 
                  y\^2)\)\ \@\(x + \@\(x\^2 + y\^2\)\) - 
            7\/3\ \((x\ \@\(x + \@\(x\^2 + y\^2\)\) - \@\(\(-x\) + \@\(x\^2 + y\^2\)\)\ Abs[
                      y])\) + \@21\ \((\@\(x\^2 + y\^2\)\ \@\(\(-x\) + \@\(x\^2 + y\^2\)\)\ Sign[y] - 
                  5\/3\ \((y\ \@\(x + \@\(x\^2 + y\^2\)\) + 
                        x\ \@\(\(-x\) + \@\(x\^2 + y\^2\)\)\ Sign[
                            y])\))\))\)\)

(10.27) Candidate Φ in cartesian coordinates

candΦ[x_, 
        y_] = 
1/
          Sqrt[193*2*
              Pi] (Sqrt[
                21] *
(Sign[y]*(x^2 + y^2)^(1/2)*
                    Sqrt[(x^2 + y^2)^(1/2) - x] - 
                  5/3*(y*Sqrt[(x^2 + y^2)^(1/2) + x] + 
                        x*Sign[y]*
                          Sqrt[(x^2 + y^2)^(1/2) - 
                              x])) - ((x^2 + y^2)^(1/2)*
                  Sqrt[(x^2 + y^2)^(1/2) + x] - 
                7/3 (x*Sqrt[(x^2 + y^2)^(1/2) + x] - 
                      Abs[y]*Sqrt[(x^2 + y^2)^(1/2) - 
                            x])))

(10.28)  Graphs of candidate W  in  a square domain : Fig.6

candW[x_, 
        y_] = 
1/
          Sqrt[193*2*
              Pi]  (Sqrt[
                21] *
(Sign[y]*(x^2 + y^2)^(1/2)*
                    Sqrt[(x^2 + y^2)^(1/2) - x] - 
                  5/3*(y*Sqrt[(x^2 + y^2)^(1/2) + x] + 
                        x*Sign[y]*
                          Sqrt[(x^2 + y^2)^(1/2) - 
                              x])) + (x^2 + y^2)*
              Sqrt[(x^2 + y^2)^(1/2) + x] - 
            7/3 (x*Sqrt[(x^2 + y^2)^(1/2) + x] - 
                  Abs[y]*Sqrt[(x^2 + y^2)^(1/2) - 
                        x]))

Plot3D[
      candW[x, y] ,  {x, -1, 1}, {y, -1, 
        1},  
PlotPoints -> 50, 
      PlotRange -> {-1, 1}, 
      ViewPoint -> {1.196,  -2.581,  1.833}]

(10.29)  Graphs of candidates Φ  in  a square domain :  Fig.9

Plot3D[
          candΦ[x, y] ,  {x, -1, 1}, {y, -1, 
            1},  
PlotPoints -> 40, 
          PlotRange -> {-1, 1}, 
          ViewPoint -> {1.196,  -2.581,  1.833}]

⁃  
            SurfaceGraphics  ⁃

(10.30)  Level lines of candidate W (cartesian coordinates): Fig. 5

ContourPlot[candW[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 140, Contours -> 30]

(10.31)  Level lines of candidate  Φ   (cartesian coordinates): Fig. 8

candΦ[x_, 
        y_] = 
1/
          Sqrt[193*2*
              Pi] (Sqrt[
                21] *
(Sign[y]*(x^2 + y^2)^(1/2)*
                    Sqrt[(x^2 + y^2)^(1/2) - x] - 
                  5/3*(y*Sqrt[(x^2 + y^2)^(1/2) + x] + 
                        x*Sign[y]*
                          Sqrt[(x^2 + y^2)^(1/2) - 
                              x])) - ((x^2 + y^2)^(1/2)*
                  Sqrt[(x^2 + y^2)^(1/2) + x] - 
                7/3 (x*Sqrt[(x^2 + y^2)^(1/2) + x] - 
                      Abs[y]*Sqrt[(x^2 + y^2)^(1/2) - 
                            x])))

ContourPlot[candΦ[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 140, Contours -> 30]

ClearAll

COMPLEX BASIS :   (H^2) complete set for V   (redundant in (L^2)) :   { {(Subscript[v, k])}  , {(Subscript[z, k])}  }   k ∈ ℤ    (indexed by k here, parametrized by h in the paper)

and scalar products  (Subscript[((.,.)), ρ])  in (H^2) (\(TraditionalForm\`\(\(B\_\(\(ρ\)\(\ \)\)\)\(\ \)\(\\)\)\) Γ)

Differential operators

dx[u_] := 
      Cos[ϑ]*D[u, r] - 
        Sin[ϑ]*D[u, ϑ]/r
dy[u_] := 
      Sin[ϑ]*D[u, r] + 
        Cos[ϑ]*D[u, ϑ]/r
dxx[u_] := dx[dx[u]]
dyy[u_] := dy[dy[u]]
dxy[u_] := dx[dy[u]]
dyx[u_] := dy[dx[u]]
lap[u_] := dxx[u] + dyy[u]
hes2[u_] := dxx[u]^2 + dyy[u]^2 + 2*dxy[u]^2
prodscalH2[u_, v_] := 
      Integrate[(dxx[u]*dxx[v] + dyy[u]*dyy[v] + 2  dxy[u]*dxy[v])*r, {r, 
          0, R}, {ϑ,  -Pi,  Pi}]

prodscalHERMITIANOH2[u_, v_] := 
      Integrate[(Simplify[dxx[u]*  Conjugate[dxx[v]]] + 
              Simplify[dyy[u]*Conjugate[dyy[v]]] + 
              Simplify[2  dxy[u]*Conjugate[dxy[v]]])*r, {r, 0, 
          R}, {ϑ,  -Pi,  Pi}, 
        Assumptions -> {k ∈ Integers, ϑ ∈ 
              Reals, r ∈ Reals}]

lap[lap[r^p*ψ[ϑ]]]

Simplify[%]

(Subscript[v, k])  ,  (Subscript[z, k])   ( (L^2))   Redundant  complete sets  (parameter h in  the paper)

(Subscript[v, k])  ,  (Subscript[z, k])   ( (H^2))   Minimal  complete sets  (parameter h in  the paper)

v[k_] := 
      r^(Abs[k] - 1/2)*E^(I*(k - 1/2)*ϑ)

z[k_] := 
      r^(Abs[k] + 3/2)*E^(I*(k - 1/2)*ϑ)

cv[k_] := 
      r^(Abs[k] - 1/2)*
        E^(-I*(k - 1/2)*ϑ)

cz[k_] := 
      r^(Abs[k] + 3/2)*
        E^(-I*(k - 1/2)*ϑ)

hes11[u_] := dxx[u]

hes22[u_] := dyy[u]

hes12[u_] := dxy[u]

hes21[u_] := dyx[u]

(10.32)  HESSIAN OF FUNCTIONS OF FIRST KIND   ((Subscript[v, k]))

hes11[v[k]]

Simplify[%,  {k} ∈ Integers ]

TeXForm[%]

hes22[v[k]]

Simplify[% ,  {k} ∈ Integers ]

TeXForm[%]

hes12[v[k]]

Simplify[%,  {k} ∈ Integers ]

TeXForm[%]

hes21[v[k]]

Simplify[ % ,  {k} ∈ Integers ]

TeXForm[%]

(10.33)  HESSIAN OF FUNCTIONS OF SECOND KIND   ((Subscript[z, k]))

hes11[z[k]]

Simplify[ % ,  {k} ∈ Integers  ]

TeXForm[%]

hes22[z[k]]

Simplify[ % ,  {k} ∈ Integers ]

TeXForm[%]

hes12[z[k]]

Simplify[ % ,  {k} ∈ Integers ]

TeXForm[%]

hes21[z[k]]

Simplify[ % ,  {k} ∈ Integers  ]

TeXForm[%]

(10.34)  COMPUTATION OF SCALAR PRODUCTS IN  (H^2)  FOR  { {(Subscript[v, k])}  , {(Subscript[z, k])}  }   k ∈ ℤ

prodscalH2[v[k], cv[h]]

Simplify[%, {h, k} ∈ Integers]

prodscalH2[v[k], cz[h]]

Simplify[%, {h, k} ∈ 
        Integers]

prodscalH2[z[k], cz[h]]

Simplify[%, {h, k} ∈ Integers]

REAL BASIS :   (H^2) complete set for V   (redundant in (L^2)) :  {  {(Subscript[f1, k])}  ,   {(Subscript[f2, k])}  ,  { (Subscript[f3, k])}  , {(Subscript[f4, k])}  ,}   k ∈ ℤ
(indexed by k here, parametrized by h in the paper)

and scalar products  (Subscript[((.,.)), ρ])  in (H^2) (\(TraditionalForm\`\(\(B\_\(\(ρ\)\(\ \)\)\)\(\ \)\(\\)\)\) Γ)

ClearAll

dx[u_] := Cos[t]*D[u, r] - Sin[t]*D[u, t]/r
dy[u_] := Sin[t]*D[u, r] + Cos[t]*D[u, t]/r
dxx[u_] := dx[dx[u]]
dyy[u_] := dy[dy[u]]
dxy[u_] := dx[dy[u]]
dyx[u_] := dx[dy[u]]
lap[u_] := dxx[u] + dyy[u]
hes2[u_] := dxx[u]^2 + dyy[u]^2 + 2*dxy[u]^2
prodscalH2[u_, v_] := 
      Integrate[(Simplify[dxx[u]]*
                Simplify[dxx[v]] + 
Simplify[dyy[u]]*
                Simplify[dyy[v]] + 2  Simplify[dxy[u]]*Simplify[dxy[v]])*
          r, 
{r, 0, R}, {t,  -Pi,  Pi}]

(Subscript[f1, k])  ,  (Subscript[f2, k]) ,  (Subscript[f3, k]),   (Subscript[f4, k])           ( (L^2))   Redundant  complete set  (parameter h in  the paper)

(Subscript[f1, k])  , (Subscript[f2, k]) ,  (Subscript[f3, k]),    (Subscript[f4, k])           ( (H^2))   Minimal  complete set  (parameter h in  the paper)

f1[k_] := r^(k + 3/2) Cos[(k + 3/2) t]

f2[k_] := r^(k + 3/2) Sin[(k + 3/2) t]

f3[k_] := r^(k + 3/2) Cos[(k - 1/2) t]

f4[k_] := r^(k + 3/2) Sin[(k - 1/2) t]

hes11[u_] := dxx[u]

hes22[u_] := dyy[u]

hes12[u_] := dxy[u]

hes21[u_] := dyx[u]

(10.35)  HESSIAN OF FUNCTIONS OF TYPE f OF FIRST KIND   ((Subscript[f1, k]))

hes11[f1[k]]

Simplify[%]

hes22[f1[k]]

Simplify[%]

hes12[f1[k]]

Simplify[%]

hes21[f1[k]]

Simplify[%]

(10.36)  HESSIAN OF FUNCTIONS OF TYPE   f  OF SECOND KIND   ((Subscript[f2, k]))

hes11[f2[k]]

Simplify[%]

hes22[f2[k]]

Simplify[%]

hes12[f2[k]]

Simplify[%]

hes21[f2[k]]

Simplify[%]

(10.37)  HESSIAN OF FUNCTIONS OF TYPE   f  OF THIRD KIND   ((Subscript[f3, k]))

hes11[f3[k]]

Simplify[%]

hes22[f3[k]]

Simplify[%]

hes12[f3[k]]

Simplify[%]

hes21[f3[k]]

Simplify[%]

(10.38) HESSIAN OF FUNCTIONS OF TYPE f OF FOURTH KIND   ((Subscript[f4, k]))

hes11[f4[k]]

Simplify[%]

hes22[f4[k]]

Simplify[%]

hes12[f4[k]]

Simplify[%]

hes21[f4[k]]

Simplify[%]

lap[lap[f1]]

lap[lap[f2]]

lap[lap[f3]]

lap[lap[f4]]

(10.39)  COMPUTATION OF SCALAR PRODUCTS IN  (H^2)  FOR  {  {(Subscript[f1, k])}  ,   {(Subscript[f2, k])}  ,  { (Subscript[f3, k])}  , {(Subscript[f4, k])}  ,}   k ∈ ℤ

prodscalH2[f1[k], f1[k]]

TeXForm[%]

prodscalH2[f1[k], f2[k]]

prodscalH2[f1[k], f4[k]]

prodscalH2[f1[k], f3[k]]

prodscalH2[f2[k], f4[k]]

prodscalH2[f2[k], f3[k]]

prodscalH2[f2[k], f2[k]]

TeXForm[
      (1/4) (1 + 
            2 k) (((3 + 2 k))^2) π (R^(1 + 2 k))]

prodscalH2[f3[k], f3[k]]

Simplify[%]

TeXForm[(((1 + 2 k) (R^(1 + 2 k)) ((-9 + 22 k - 12 (k^2) + 8 (k^3)) π - 8 Sin[2 k π]))/(-4 + 8 k))]

prodscalH2[f4[k], f4[k]]

Simplify[%]

TeXForm[(((1 + 2 k) (R^(1 + 2 k)) ((-9 + 22 k - 12 (k^2) + 8 (k^3)) π + 8 Sin[2 k π]))/(-4 + 8 k))]

Factor[-9 + 22  k - 12  k^2 + 8  k^3]

PAY ATTENTION :  
        from now non h labels a generic index different from k (not the label of f1,  f2,  ,  ... )
Sin (h - k) 
      Pi  always stans for 0 since 
h and k are integers.

prodscalH2[f1[k], f1[h]]

Simplify[%]

prodscalH2[f2[k], f2[h]]

Simplify[%]

prodscalH2[f3[k], f3[h]]

Simplify[%]

prodscalH2[f4[k], f4[h]]

Simplify[%]

prodscalH2[f1[k], f2[h]]

prodscalH2[f1[k], f3[h]]

Simplify[%]

prodscalH2[f1[k], f4[h]]

prodscalH2[f2[k], f4[h]]

Simplify[%]

prodscalH2[f3[k], f4[h]]

dxx[f1[k]]

Simplify[%]

TeXForm[
      (1/4) (3 + 8 k + 4 (k^2)) (r^(-(1/2) + k)) Cos[
          (1/2) (t - 2 k t)]]

dxy[f1[k]]

Simplify[%]

TeXForm[
      (1/4) (3 + 8 k + 4 (k^2)) (r^(-(1/2) + k)) Sin[
          (1/2) (t - 2 k t)]]

dyy[f1[k]]

Simplify[%]

TeXForm[-(1/4) (3 + 8 k + 
            4 (k^2)) (r^(-(1/2) + k)) Cos[
          (1/2) (t - 2 k t)]]

dxx[f2[k]]

Simplify[%]

TeXForm[-(1/4) (3 + 8 k + 
            4 (k^2)) (r^(-(1/2) + k)) Sin[
          (1/2) (t - 2 k t)]]

dxy[f2[k]]

Simplify[%]

TeXForm[
      (1/4) (3 + 8 k + 4 (k^2)) (r^(-(1/2) + k)) Cos[
          (1/2) (t - 2 k t)]]

dyy[f2[k]]

Simplify[%]

TeXForm[
      (1/4) (3 + 8 k + 4 (k^2)) (r^(-(1/2) + k)) Sin[
          (1/2) (t - 2 k t)]]

dxx[f3[k]]

Simplify[%]

TeXForm[
      (1/4) (1 + 
            2 k) (r^(-(1/2) + k)) ((-1 + 
                  2 k) Cos[((5/2) - k) t] + 
            4 Cos[(-(1/2) + k) t])]

dxy[f3[k]]

Simplify[%]

TeXForm[
      (1/4) (-1 + 2 k) (1 + 
            2 k) (r^(-(1/2) + k)) Sin[((5/2) - 
                k) t]]

dyy[f3[k]]

Simplify[%]

TeXForm[-(1/4) (1 + 
            2 k) (r^(-(1/2) + k)) ((-1 + 
                  2 k) Cos[((5/2) - k) t] - 
            4 Cos[(-(1/2) + k) t])]

dxx[f4[k]]

Simplify[%]

TeXForm[
      (1/4) (1 + 
            2 k) (r^(-(1/2) + k)) ((1 - 
                  2 k) Sin[((5/2) - k) t] + 
            4 Sin[(-(1/2) + k) t])]

dxy[f4[k]]

Simplify[%]

TeXForm[
      (1/4) (-1 + 2 k) (1 + 
            2 k) (r^(-(1/2) + k)) Cos[((5/2) - 
                k) t]]

dyy[f4[k]]

Simplify[%]

TeXForm[
      (1/4) (1 + 
            2 k) (r^(-(1/2) + k)) ((-1 + 
                  2 k) Sin[((5/2) - k) t] + 
            4 Sin[(-(1/2) + k) t])]

(10.40),(10.41)  Evaluation of squared hessian jump for candidates W , Φ   along jump set

dx[u_] := Cos[t]*D[u, r] - Sin[t]*D[u, t]/r
dy[u_] := Sin[t]*D[u, r] + Cos[t]*D[u, t]/r
dxx[u_] := dx[dx[u]]
dyy[u_] := dy[dy[u]]
dxy[u_] := dx[dy[u]]
dyx[u_] := dy[dx[u]]
lap[u_] := dxx[u] + dyy[u]
hes2[u_] := dxx[u]^2 + dyy[u]^2 + 2*dxy[u]^2

W[r_, 
        t_]  :=   r^(3/2) *  Sqrt[ α  / ( 
              193 * Pi) ] * 
                                          (   Sqrt[
                21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )  +  
                                                                   (  Cos[t/2])    -  
            7/3 * Cos[
                3 * t/
                    2]        )

Φ[r_, 
        t_]  :=   r^(3/
              2) * Sqrt[ α  / ( 
              193 * Pi) ] * 
                                          (   Sqrt[
                21] * (  Sin[t/2]  -  
                  5/3  * 
                    Sin[ 3 * 
                        t/2] )   -  
                                                                   (  Cos[t/2]    -  
                7/3 * Cos[
                    3 * t/2] )       )

Simplify[hes2[W[r, π]]]

Simplify[hes2[W[r, -π]]]

Simplify[hes2[Φ[r, π]]]

Simplify[hes2[Φ[r, -π]]]

How to evaluate modes and their derivatives and hessian along jump set

u /. t -> -Pi

u /. t -> Pi

Simplify[dy[u]] /. t -> Pi

Simplify[dy[u]] /. t -> -Pi

Simplify[dy[dy[u]]] /. t -> -Pi

Simplify[dy[dy[u]]] /. t -> Pi

Simplify[dy[dy[dy[u]]] + 2*dx[dx[dy[u]]]] /. t -> Pi

Simplify[dy[dy[dy[u]]] + 2*dx[dx[dy[u]]]] /. t -> -Pi

Simplify[hes2[u]] /. t -> Pi

Simplify[hes2[u]] /. t -> -Pi

About  Mumford-Shah functional

u := r^(1/2)*Sin[t/2]

Simplify[lp[u]]

Simplify[dy[u]] /. t -> -Pi

Simplify[dy[u]] /. t -> Pi

(dx[u]^2 + dy[u]^2)*r

Integrate[%, {r, 0, 1}, {t, -Pi, Pi}]

\(Simplify[\((dx[u]^2 + dy[u]^2)\)*Cos[t] - 
2*dx[u]*D[u, r]]\)

Integrte[r*%, {t, -Pi, Pi}]

\(Simplify[\((dx[u]^2 + dy[u]^2)\)*\((dx[φ] + dy[ψ])\) - 
        2*
		\((dx[
                  u]*\((dx[u]*dx[φ] + 
                    dy[u]*dx[ψ])\) + 
				dy[
                  u]*\((dx[u]*dy[φ] + 
                    dy[u]*dy[ψ])\))\)]\)

Integrate[%*r, {t, -Pi, Pi}, {r, 0, 1}]

ParametricPlot3D[{r*Cos[t], r*Sin[t], u}, {r, 
          0, 1}, {t, -Pi, Pi}, AspectRtio -> 1/2, 
        ViewPoint -> {1.051,  -3.180,  1.105}]

Graph of candidate for Mumford Shah functional

candMS[s_, t_] := 
      If[x > 0, Im[Sqrt[2  (x + I y)/Pi]] , 
          Im[Sqrt[2  (x + I y)/Pi]] ] /. {x -> s, 
          y -> t}

Plot3D[candMS[x, y], {x, -1, 1}, {y, -1, 1}, 
      PlotPoints -> 40, PlotRange -> {-1.5, 1.5}, 
      ViewPoint -> {1.196,  -2.581,  1.833}]

Computations for generic crack-tip conditions  (generic vector fields η in Theorem 5.5)

φ := (8*r*Cos[t] + 1)*(1 - r^2)^6

ψ := (r*Cos[t] + 1)*(1 - r^2)^6

φ := (1 - r^2)^6

ψ = 0

\(Simplify[\(-\((hes2[σ]*r*Cos[t] + 
                2*
		dx[σ]*\((dy[lp[σ]]*r*Sin[t] + 
                      dx[lp[σ]]*r*Cos[t])\)
		 - 
                2*r*Cos[t]*\((dxx[σ]^2 + dxy[σ]^2)\) - 
                2*r*Sin[
                    t]*\((dxx[σ]*dxy[σ] + 
                      dxy[σ]*dyy[σ])\))\)\)*φ + 

        2*dx[σ]*\((\((dxx[σ]*r*Cos[t] + 
                    dxy[σ]*r*Sin[t])\)*
                dx[φ] + \((dxy[σ]*r*Cos[t] + 
                    dyy[σ]*r*Sin[t])\)*dy[φ])\)]\)

Integrate[%, {t, -Pi, Pi}]

% /. r -> 0

Simplify[%]

σ = r^4*Cos[4*t] - r^2

Global Euler equation  (Theorem 5.1 without g,  test functions  φ  e  ψ  are needed)    
Symbolic computations about Section 5

φ := (8*r*Cos[t] + 1)*(R^2 - r^2)^3

ψ := (r*Cos[t] + 1)*(R^2 - r^2)^3

φ := (1 - r^2)^6

ψ := 0

φ

ψ

\(Integrte[\ 
      r*Simplify[
          hes2[ω]*\((dx[φ] + dy[ψ])\)
		 - 
            4*
		dxy[ω]*\((dxx[ω]*dy[φ] + 
                  dxy[ω]*dx[φ] + 
				dx[ω]*
                    dxy[φ] + 
                  dxy[ω]*dy[ψ] + 
				dyy[ω]*
                    dx[ψ] + dy[ω]*dxy[ψ])\)
		 - 
            2*
		dxx[ω]*\((dxx[ω]*dx[φ] + 
                  dxx[ω]*dx[φ] + 
				dx[ω]*
                    dxx[φ] + dxy[ω]*dx[ψ] + 
                  dxy[ω]*dx[ψ] + 
                  dy[ω]*dxx[ψ])\)
		 - 
            2*
		dyy[ω]*\((dxy[ω]*dy[φ] + 
                  dxy[ω]*dy[φ] + 
				dx[ω]*
                    dyy[φ] + dyy[ω]*dy[ψ] + 
                  dyy[ω]*dy[ψ] + 
                  dy[ω]*dyy[ψ])\)
		], {t, \(-Pi\), Pi}, {r, 
        0, R}]\)

dx[φ]

Integrte[dx[φ], {r, 0, R}] /. t -> Pi

ω

Simplify[%]

Integration by parts  (Theorem 5.2 without g,   test functions  φ  e  ψ  are needed ) 
Symbolic computations about Section 5

u = r^4*Cos[t]^4 + r^4*Cos[t]^2*Sin[t]^2

ω = u

Simplify[lp[u]]

Simplify[lp[%]]

φ = r^2*(Cos[t]^2) + r^3*Sin[t]^3

ψ = (r*Cos[t] + 1)*(1 + r^2)

φ := (1 - r^2)^6

ψ := 0

\(Simplify[
      hes2[ω]*\((dx[φ] + dy[ψ])\)
		 - 
        4*
		dxy[ω]*\((dxx[ω]*dy[φ] + 
              dxy[ω]*dx[φ] + 
				dx[ω]*
                dxy[φ] + 
              dxy[ω]*dy[ψ] + 
				dyy[ω]*dx[ψ] + 
              dy[ω]*dxy[ψ])\)
		 - 
        2*
		dxx[ω]*\((dxx[ω]*dx[φ] + 
              dxx[ω]*dx[φ] + 
				dx[ω]*
                dxx[φ] + dxy[ω]*dx[ψ] + 
              dxy[ω]*dx[ψ] + dy[ω]*dxx[ψ])\)
		 - 
        2*
		dyy[ω]*\((dxy[ω]*dy[φ] + 
              dxy[ω]*dy[φ] + 
				dx[ω]*
                dyy[φ] + dyy[ω]*dy[ψ] + 
              dyy[ω]*dy[ψ] + 
              dy[ω]*dyy[ψ])\)
		]\)

Integrte[ %*r, {r, 0, R}, {t, -Pi, Pi}]

-2*
      Simplify[lp[lp[u]]*(φ*dx[u] + ψ*dy[u])]

1 = Integrte[ %*r, {t, -Pi, Pi}, {r, 0, 1}]

Simplify[
      hes2[u]*(φ*Cos[t] + ψ*Sin[t])  - 
        2*φ*(dxx[u]*dxx[u]*Cos[t] + dxx[u]*dxy[u]*Sin[t] + 
              dxy[u]*dxy[u]*Cos[t] + dxy[u]*dyy[u]*Sin[t]) - 
        2*ψ*(dxx[u]*dxy[u]*Cos[t] + dxy[u]*dxy[u]*Sin[t] + 
              dxy[u]*dyy[u]*Cos[t] + dyy[u]*dyy[u]*Sin[t]) + 
        2*φ*dx[u]*(dx[lp[u]]*Cos[t] + dy[lp[u]]*Sin[t]) + 
        2*ψ*dy[u]*(dx[lp[u]]*Cos[t] + dy[lp[u]]*Sin[t])]

2 = Integrte[%, {t, -Pi, Pi}]

Simplify[-2*dx[u]*
          dx[φ]*(dxx[u]*Cos[t] + dxy[u]*Sin[t]) - 
        2*dy[φ]*dx[u]*(dxy[u]*Cos[t] + dyy[u]*Sin[t]) - 
        2*dy[u]*dx[ψ]*(dxx[u]*Cos[t] + dxy[u]*Sin[t]) - 
        2*dy[ψ]*dy[u]*(dxy[u]*Cos[t] + dyy[u]*Sin[t])]

3 = Integrte[%, {t, -Pi, Pi}]

Integration by parts over half-disk  with generic vector field  η    
(Theorem 5.2 without g,  test functions  φ  e  ψ  are needed )

u = r^4*Cos[t]^4 - 2*r^3*Cos[t]^2*Sin[t]

u = r^4*Cos[t]^4 + r^4*Cos[t]^2*Sin[t]^2

ω = u

Simplify[lp[u]]

Simplify[lp[%]]

φ = r^2*(Cos[t]^2) - 3*r^3*Sin[t]^3

ψ = (r*Cos[t] + 1)*(1 + r^2)

φ := (1 - r^2)^6

ψ = r^2 - 7*r^5*Sin[t]^5

\(LHS = 
      Integrte[\ 
        r*Simplify[
            hes2[ω]*\((dx[φ] + dy[ψ])\)
		 - 
              4*
		dxy[ω]*\((dxx[ω]*dy[φ] + 
                    dxy[ω]*dx[φ] + 
				dx[ω]*
                      dxy[φ] + 
                    dxy[ω]*dy[ψ] + 
				dyy[ω]*
                      dx[ψ] + dy[ω]*dxy[ψ])\)
		 - 
              2*
		dxx[ω]*\((dxx[ω]*dx[φ] + 
                    dxx[ω]*dx[φ] + 
				dx[ω]*
                      dxx[φ] + dxy[ω]*dx[ψ] + 
                    dxy[ω]*dx[ψ] + 
                    dy[ω]*dxx[ψ])\)
		 - 
              2*
		dyy[ω]*\((dxy[ω]*dy[φ] + 
                    dxy[ω]*dy[φ] + 
				dx[ω]*
                      dyy[φ] + dyy[ω]*dy[ψ] + 
                    dyy[ω]*dy[ψ] + 
                    dy[ω]*dyy[ψ])\)
		], {r, 0, R}, {t, 0, 
          Pi}]\)

1 = 
      Integrte[ -2*
          Simplify[lp[lp[u]]*(φ*dx[u] + ψ*dy[u])]*r, {r, 
          0, R}, {t, 0, Pi}]

2 = 
      R*Integrte[
            Simplify[
              hes2[u]*(φ*Cos[t] + ψ*Sin[t])  - 
                2*φ*(dxx[u]*dxx[u]*Cos[t] + 
                      dxx[u]*dxy[u]*Sin[t] + dxy[u]*dxy[u]*Cos[t] + 
                      dxy[u]*dyy[u]*Sin[t]) - 
                2*ψ*(dxx[u]*dxy[u]*Cos[t] + dxy[u]*dxy[u]*Sin[t] + 
                      dxy[u]*dyy[u]*Cos[t] + dyy[u]*dyy[u]*Sin[t]) + 
                2*φ*
                  dx[u]*(dx[lp[u]]*Cos[t] + dy[lp[u]]*Sin[t]) + 
                2*ψ*
                  dy[u]*(dx[lp[u]]*Cos[t] + dy[lp[u]]*Sin[t])], {t, 0, 
              Pi}] /. r -> R

3 = 
      Integrte[Simplify[
            hes2[u]*(-ψ)  - 
              2*φ*(-dxx[u]*dxy[u] - dxy[u]*dyy[u]) - 
              2*ψ*(-dxy[u]*dxy[u] - dyy[u]*dyy[u]) + 
              2*φ*dx[u]*(-dy[lp[u]]) + 
              2*ψ*dy[u]*(-dy[lp[u]])], {r, 0, R}] /. 
        t -> 0

4 = 
      Integrte[Simplify[
            hes2[u]*(-ψ)  - 
              2*φ*(-dxx[u]*dxy[u] - dxy[u]*dyy[u]) - 
              2*ψ*(-dxy[u]*dxy[u] - dyy[u]*dyy[u]) + 
              2*φ*dx[u]*(-dy[lp[u]]) + 
              2*ψ*dy[u]*(-dy[lp[u]])], {r, 0, R}] /. 
        t -> Pi

5 = 
      R*Integrte[
            Simplify[-2*dx[u]*
                  dx[φ]*(dxx[u]*Cos[t] + dxy[u]*Sin[t]) - 
                2*dy[φ]*dx[u]*(dxy[u]*Cos[t] + dyy[u]*Sin[t]) - 
                2*dy[u]*dx[ψ]*(dxx[u]*Cos[t] + dxy[u]*Sin[t]) - 
                2*dy[ψ]*dy[u]*(dxy[u]*Cos[t] + dyy[u]*Sin[t])], {t, 
              0, Pi}] /. r -> R

6 = 
      Integrte[Simplify[-2*dx[u]*dx[φ]*(-dxy[u]) - 
              2*dy[φ]*dx[u]*(-dyy[u]) - 
              2*dy[u]*dx[ψ]*(-dxy[u]) - 
              2*dy[ψ]*dy[u]*(-dyy[u])], {r, 0, R}] /. 
        t -> 0

7 = 
      Integrte[Simplify[-2*dx[u]*dx[φ]*(-dxy[u]) - 
              2*dy[φ]*dx[u]*(-dyy[u]) - 
              2*dy[u]*dx[ψ]*(-dxy[u]) - 
              2*dy[ψ]*dy[u]*(-dyy[u])], {r, 0, R}] /. 
        t -> Pi

RHS = Simplify[1 + 2 + 3 + 4 + 5 + 6 + 7]

Simplify[LHS - RHS]