(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, nohscroll, noKeepOnOnePage, preserveAspect, cellOutline, groupLikeTitle, center, M7, R65535, L1, e8, 24, "CalcMath"; ; fontset = subtitle, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeTitle, center, M7, L1, e6, 18, "CalcMath"; ; fontset = subsubtitle, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeTitle, center, M7, L1, e6, 18, "CalcMath"; ; fontset = section, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, grayBox, M22, L1, a20, 18, "CalcMath"; ; fontset = subsection, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, blackBox, M19, R39320, G26214, B13107, L1, a15, 16, "CalcMath"; ; fontset = subsubsection, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, whiteBox, M18, L1, a12, 14, "CalcMath"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, B65535, L1, 14, "CalcMath"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "CalcMath"; ; fontset = input, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, preserveAspect, groupLikeOutput, M42, N23, R65535, B65535, L1, 12, "Courier"; ; fontset = print, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, nowordwrap, preserveAspect, groupLikeGraphics, M7, l141, w248, h323, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = leftheader, nohscroll, cellOutline, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = leftfooter, cellOutline, blackBox, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M7, G65535, L1, 14, "Times"; ; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, R32767, B32767, L1, 12, "Courier"; ; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, right, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; showRuler; magnification = 125; currentKernel; ] :[font = title; inactive; preserveAspect; rightWrapOffset = 427] Construction of a certain n-superharmonic function :[font = subtitle; inactive; preserveAspect] Juan J. Manfredi and Enrique Villamor :[font = special2; inactive; preserveAspect] Appendix to the paper: "Mappings with Integrabe Dilatation in Higher Dimensions" :[font = subsection; inactive; preserveAspect; startGroup] Introduction :[font = text; inactive; preserveAspect] Let a be a small positive parameter satisfying 00. 4. log(Œ’Ú) É fŒ (r) É log(Œ’Ú) + ª’Ú + log 2 for r< eÑä. 5. fŒ (y)=log(´’Ú) for a É |y| É eÑä. 6. The vector field |„fŒ (y)|öÑÛ„fŒ (y) is of class CÚ. :[font = text; inactive; preserveAspect; endGroup] As a approaches zero fŒ (y) approximates to log(´’Ú), but it remains bounded (by log(Œ’Ú) + ª’Ú + log 2) and smooth. :[font = subsection; inactive; preserveAspect; startGroup] The Construction :[font = text; inactive; preserveAspect] We start by considering u(r)=log(¨’Ú) which is certainly n-superharmonic. :[font = input; preserveAspect; startGroup] u[r_]=Log[1/r] :[font = output; output; inactive; preserveAspect; endGroup] Log[r^(-1)] ;[o] 1 Log[-] r :[font = text; inactive; preserveAspect] Fix a>0. The next function agrees with u up to second order at r=a and it has negative n-Laplacian. However, it is not smooth at zero :[font = input; preserveAspect; startGroup] v[r_,a_]=u[a]+u'[a] (r-a)+(1/2)u''[a] (r-a)^2 :[font = output; output; inactive; preserveAspect; endGroup] -((-a + r)/a) + (-a + r)^2/(2*a^2) + Log[a^(-1)] ;[o] 2 -a + r (-a + r) 1 -(------) + --------- + Log[-] a 2 a 2 a :[font = text; inactive; preserveAspect] We will use v for a/2a/2 eq3=w''[a/2]==D[v[r,a],{r,2}]/.r->a/2 eq4=w[0]==v[a,a/2]+1 :[font = output; output; inactive; preserveAspect] b + (a^2*c)/4 + (a^4*d)/16 + (a^6*e)/64 == 5/8 + Log[a^(-1)] ;[o] 2 4 6 a c a d a e 5 1 b + ---- + ---- + ---- == - + Log[-] 4 16 64 8 a :[font = output; output; inactive; preserveAspect] a*c + (a^3*d)/2 + (3*a^5*e)/16 == -3/(2*a) ;[o] 3 5 a d 3 a e -3 a c + ---- + ------ == --- 2 16 2 a :[font = output; output; inactive; preserveAspect] 2*c + 3*a^2*d + (15*a^4*e)/8 == a^(-2) ;[o] 4 2 15 a e -2 2 c + 3 a d + ------- == a 8 :[font = output; output; inactive; preserveAspect; endGroup] b == 1/2 + Log[2/a] ;[o] 1 2 b == - + Log[-] 2 a :[font = text; inactive; preserveAspect] Note that the last equation is not a missprint. It has to be v[a,a/2] and not v[a/2,a] for our argument to work. :[font = text; inactive; preserveAspect] Now we let Mathematica work ... (with a little help from us) :[font = input; preserveAspect; startGroup] sols=InputForm[Simplify[Solve[ {eq1,eq2,eq3,eq4},{b,c,d,e}] ]] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] InputForm[{{b -> 1/2 + Log[2/a], c -> (5 + 12*Log[a^(-1)] - 12*Log[2/a])/a^2, d -> (4*(-7 - 12*Log[a^(-1)] + 12*Log[2/a]))/a^4, e -> (8*(5 + 8*Log[a^(-1)] - 8*Log[2/a]))/a^6}}] ;[o] {{b -> 1/2 + Log[2/a], c -> (5 + 12*Log[a^(-1)] - 12*Log[2/a])/a^2, d -> (4*(-7 - 12*Log[a^(-1)] + 12*Log[2/a]))/a^4, e -> (8*(5 + 8*Log[a^(-1)] - 8*Log[2/a]))/a^6}} :[font = text; inactive; preserveAspect] We rename w to ww in order to incorporate the dependence on a: :[font = input; preserveAspect; startGroup] ww[r_,a_]=InputForm[w[r]/.sols[[1]]] :[font = output; output; inactive; preserveAspect; fontLeading = 0; endGroup] InputForm[{1/2 + (r^2*(5 + 12*Log[a^(-1)] - 12*Log[2/a]))/a^2 + (8*r^6*(5 + 8*Log[a^(-1)] - 8*Log[2/a]))/a^6 + Log[2/a] + (4*r^4*(-7 - 12*Log[a^(-1)] + 12*Log[2/a]))/a^4}] ;[o] {1/2 + (r^2*(5 + 12*Log[a^(-1)] - 12*Log[2/a]))/a^2 + (8*r^6*(5 + 8*Log[a^(-1)] - 8*Log[2/a]))/a^6 + Log[2/a] + (4*r^4*(-7 - 12*Log[a^(-1)] + 12*Log[2/a]))/a^4} :[font = input; preserveAspect] :[font = text; inactive; preserveAspect] Let us simplify aÑÚ to Œ’Ú and rename again: :[font = input; preserveAspect; startGroup] www[r_,a_]=(1/2+Log[2/a] +((5-12Log[2])/a^2)r^2 +4((-7+12Log[2])/a^4)r^4 +8((5-8Log[2])/a^6)r^6) ;[o] 2 4 6 1 2 (5 - 12 Log[2]) r 4 (-7 + 12 Log[2]) r 8 (5 - 8 Log[2]) r www[r_, a_] = - + Log[-] + ------------------ + --------------------- + ------------------- 2 a 2 4 6 a a a :[font = output; output; inactive; preserveAspect; endGroup] 1/2 + (r^2*(5 - 12*Log[2]))/a^2 + (8*r^6*(5 - 8*Log[2]))/a^6 + (4*r^4*(-7 + 12*Log[2]))/a^4 + Log[2/a] ;[o] 2 6 1 r (5 - 12 Log[2]) 8 r (5 - 8 Log[2]) - + ------------------ + ------------------- + 2 2 6 a a 4 4 r (-7 + 12 Log[2]) 2 --------------------- + Log[-] 4 a a :[font = text; inactive; preserveAspect] We now piece together the funcitons u,v and www: :[font = input; preserveAspect] j[r_,a_]:=Which[ r<=a/2,www[r,a], a/2a,u[r] ] :[font = text; inactive; preserveAspect] Here comes our first picture. The thicker graph is the function j(r,.1): :[font = input; preserveAspect; startGroup] Plot[{j[r,.1],www[r,.1],v[r,.1],u[r]},{r,0.001,.2}, {PlotStyle->{Thickness[.006],Thickness[.001], Thickness[.001],Thickness[.001]}},PlotPoints->200] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 71; pictureWidth = 347; pictureHeight = 214; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 4.761905 0.05648 0.079163 [ [(0.05)] .2619 .05648 0 2 Msboxa [(0.1)] .5 .05648 0 2 Msboxa [(0.15)] .7381 .05648 0 2 Msboxa [(0.2)] .97619 .05648 0 2 Msboxa [(1)] .01131 .13564 1 0 Msboxa [(2)] .01131 .21481 1 0 Msboxa [(3)] .01131 .29397 1 0 Msboxa [(4)] .01131 .37313 1 0 Msboxa [(5)] .01131 .4523 1 0 Msboxa [(6)] .01131 .53146 1 0 Msboxa [(7)] .01131 .61062 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 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L .52857 .23415 L .53333 .23341 L .5381 .23268 L .54286 .23196 L .54762 .23124 L .55238 .23053 L .55714 .22983 L .5619 .22914 L .56667 .22845 L .57143 .22778 L .57619 .22711 L .58095 .22645 L .58571 .22579 L .59048 .22515 L .59524 .22451 L .6 .22388 L .60476 .22326 L .60952 .22265 L .61429 .22204 L .61905 .22144 L .62381 .22085 L .62857 .22027 L .63333 .2197 L .6381 .21913 L .64286 .21857 L .64762 .21802 L .65238 .21748 L .65714 .21695 L .6619 .21642 L .66667 .2159 L .67143 .21539 L .67619 .21489 L .68095 .21439 L .68571 .21391 L .69048 .21343 L .69524 .21296 L .7 .21249 L .70476 .21204 L .70952 .21159 L .71429 .21115 L .71905 .21072 L .72381 .2103 L .72857 .20988 L .73333 .20947 L .7381 .20907 L Mistroke .74286 .20868 L .74762 .2083 L .75238 .20792 L .75714 .20755 L .7619 .20719 L .76667 .20684 L .77143 .2065 L .77619 .20616 L .78095 .20583 L .78571 .20551 L .79048 .2052 L .79524 .20489 L .8 .2046 L .80476 .20431 L .80952 .20403 L .81429 .20375 L .81905 .20349 L .82381 .20323 L .82857 .20298 L .83333 .20274 L .8381 .20251 L .84286 .20228 L .84762 .20206 L .85238 .20185 L .85714 .20165 L .8619 .20146 L .86667 .20127 L .87143 .20109 L .87619 .20092 L .88095 .20076 L .88571 .20061 L .89048 .20046 L .89524 .20032 L .9 .20019 L .90476 .20007 L .90952 .19995 L .91429 .19985 L .91905 .19975 L .92381 .19966 L .92857 .19957 L .93333 .1995 L .9381 .19943 L .94286 .19937 L .94762 .19932 L .95238 .19928 L .95714 .19924 L .9619 .19921 L .96429 .1992 L .96667 .19919 L .96905 .19919 L Mistroke .97024 .19918 L .97143 .19918 L .97262 .19918 L .97321 .19918 L .97381 .19918 L .9744 .19918 L .9747 .19918 L .975 .19918 L .9753 .19918 L .97545 .19918 L .9756 .19918 L .97574 .19918 L .97589 .19918 L .97604 .19918 L .97619 .19918 L Mfstroke P P p p .001 w .02857 .60332 m .03095 .57122 L .03333 .54845 L .0381 .51635 L .04286 .49358 L .04762 .47591 L .05238 .46148 L .05714 .44927 L .0619 .4387 L .06667 .42938 L .07143 .42104 L .07619 .41349 L .08095 .40661 L .08571 .40027 L .09048 .3944 L .09524 .38894 L .1 .38383 L .10476 .37903 L .10952 .37451 L .11429 .37023 L .11905 .36617 L .12381 .36231 L .12857 .35862 L .13333 .3551 L .1381 .35173 L .14286 .3485 L .14762 .3454 L .15238 .34241 L .15714 .33953 L .1619 .33675 L .16667 .33407 L .17143 .33147 L .17619 .32896 L .18095 .32652 L .18571 .32416 L .19048 .32187 L .19524 .31964 L .2 .31747 L .20476 .31536 L .20952 .3133 L .21429 .3113 L .21905 .30934 L .22381 .30743 L .22857 .30557 L .23333 .30375 L .2381 .30197 L .24286 .30023 L .24762 .29853 L .25238 .29686 L .25714 .29523 L Mistroke .2619 .29363 L .26667 .29206 L .27143 .29053 L .27619 .28902 L .28095 .28754 L .28571 .28609 L .29048 .28466 L .29524 .28326 L .3 .28188 L .30476 .28053 L .30952 .2792 L .31429 .27789 L .31905 .2766 L .32381 .27534 L .32857 .27409 L .33333 .27286 L .3381 .27165 L .34286 .27046 L .34762 .26929 L .35238 .26813 L .35714 .267 L .3619 .26587 L .36667 .26477 L .37143 .26367 L .37619 .2626 L .38095 .26153 L .38571 .26049 L .39048 .25945 L .39524 .25843 L .4 .25742 L .40476 .25642 L .40952 .25544 L .41429 .25447 L .41905 .25351 L .42381 .25256 L .42857 .25163 L .43333 .2507 L .4381 .24978 L .44286 .24888 L .44762 .24799 L .45238 .2471 L .45714 .24623 L .4619 .24536 L .46667 .2445 L .47143 .24366 L .47619 .24282 L .48095 .24199 L .48571 .24117 L .49048 .24036 L .49524 .23956 L Mistroke .5 .23876 L .50476 .23797 L .50952 .23719 L .51429 .23642 L .51905 .23566 L .52381 .2349 L .52857 .23415 L .53333 .2334 L .5381 .23267 L .54286 .23194 L .54762 .23121 L .55238 .2305 L .55714 .22979 L .5619 .22908 L .56667 .22839 L .57143 .2277 L .57619 .22701 L .58095 .22633 L .58571 .22566 L .59048 .22499 L .59524 .22433 L .6 .22367 L .60476 .22302 L .60952 .22237 L .61429 .22173 L .61905 .2211 L .62381 .22046 L .62857 .21984 L .63333 .21922 L .6381 .2186 L .64286 .21799 L .64762 .21738 L .65238 .21678 L .65714 .21618 L .6619 .21559 L .66667 .215 L .67143 .21442 L .67619 .21384 L .68095 .21326 L .68571 .21269 L .69048 .21212 L .69524 .21156 L .7 .211 L .70476 .21045 L .70952 .20989 L .71429 .20935 L .71905 .2088 L .72381 .20826 L .72857 .20772 L .73333 .20719 L Mistroke .7381 .20666 L .74286 .20614 L .74762 .20561 L .75238 .20509 L .75714 .20458 L .7619 .20407 L .76667 .20356 L .77143 .20305 L .77619 .20255 L .78095 .20205 L .78571 .20155 L .79048 .20106 L .79524 .20057 L .8 .20008 L .80476 .1996 L .80952 .19912 L .81429 .19864 L .81905 .19816 L .82381 .19769 L .82857 .19722 L .83333 .19675 L .8381 .19629 L .84286 .19583 L .84762 .19537 L .85238 .19491 L .85714 .19446 L .8619 .19401 L .86667 .19356 L .87143 .19311 L .87619 .19267 L .88095 .19223 L .88571 .19179 L .89048 .19135 L .89524 .19092 L .9 .19049 L .90476 .19006 L .90952 .18963 L .91429 .18921 L .91905 .18879 L .92381 .18837 L .92857 .18795 L .93333 .18753 L .9381 .18712 L .94286 .18671 L .94762 .1863 L .95238 .18589 L .95714 .18549 L .9619 .18508 L .96667 .18468 L .97143 .18429 L Mistroke .97619 .18389 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect] We now compute and plot the first derivative of j[r,a], to make sure it negative. :[font = input; preserveAspect] wwwprima[r_,a_]=D[www[r,a],r]; vprima[r_,a_]=D[v[r,a],r]; :[font = input; preserveAspect] jprima[r_,a_]:=Which[ r<=a/2,wwwprima[r,a], a/2a,u'[r]] :[font = input; preserveAspect; pictureLeft = 141; pictureWidth = 248; pictureHeight = 153; startGroup] Plot[jprima[r,.1],{r,0,.1}, PlotRange->All] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 72; pictureTop = 2; pictureWidth = 365; pictureHeight = 225] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 9.52381 0.603319 0.037412 [ [(0.02)] .21429 .60332 0 2 Msboxa [(0.04)] .40476 .60332 0 2 Msboxa [(0.06)] .59524 .60332 0 2 Msboxa [(0.08)] .78571 .60332 0 2 Msboxa [(0.1)] .97619 .60332 0 2 Msboxa [(-15)] .01131 .04214 1 0 Msboxa [(-12.5)] .01131 .13567 1 0 Msboxa [(-10)] .01131 .2292 1 0 Msboxa [(-7.5)] .01131 .32273 1 0 Msboxa [(-5)] .01131 .41626 1 0 Msboxa [(-2.5)] .01131 .50979 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .21429 .60332 m .21429 .60957 L s P [(0.02)] .21429 .60332 0 2 Mshowa p .002 w .40476 .60332 m .40476 .60957 L s P [(0.04)] .40476 .60332 0 2 Mshowa p .002 w .59524 .60332 m .59524 .60957 L s P [(0.06)] .59524 .60332 0 2 Mshowa p .002 w .78571 .60332 m .78571 .60957 L s P [(0.08)] .78571 .60332 0 2 Mshowa p .002 w .97619 .60332 m .97619 .60957 L s P [(0.1)] .97619 .60332 0 2 Mshowa p .001 w .0619 .60332 m .0619 .60707 L s P p .001 w .1 .60332 m .1 .60707 L s P p .001 w .1381 .60332 m .1381 .60707 L s P p .001 w .17619 .60332 m .17619 .60707 L s P p .001 w .25238 .60332 m .25238 .60707 L s P p .001 w .29048 .60332 m .29048 .60707 L s P p .001 w .32857 .60332 m .32857 .60707 L s P p .001 w .36667 .60332 m .36667 .60707 L s P p .001 w .44286 .60332 m .44286 .60707 L s P p .001 w .48095 .60332 m .48095 .60707 L s P p .001 w .51905 .60332 m .51905 .60707 L s P p .001 w .55714 .60332 m .55714 .60707 L s P p .001 w .63333 .60332 m .63333 .60707 L s P p .001 w .67143 .60332 m .67143 .60707 L s P p .001 w .70952 .60332 m .70952 .60707 L s P p .001 w .74762 .60332 m .74762 .60707 L s P p .001 w .82381 .60332 m .82381 .60707 L s P p .001 w .8619 .60332 m .8619 .60707 L s P p .001 w .9 .60332 m .9 .60707 L s P p .001 w .9381 .60332 m .9381 .60707 L s P p .002 w 0 .60332 m 1 .60332 L s P p .002 w .02381 .04214 m .03006 .04214 L s P [(-15)] .01131 .04214 1 0 Mshowa p .002 w .02381 .13567 m .03006 .13567 L s P [(-12.5)] .01131 .13567 1 0 Mshowa p .002 w .02381 .2292 m .03006 .2292 L s P [(-10)] .01131 .2292 1 0 Mshowa p .002 w .02381 .32273 m .03006 .32273 L s P [(-7.5)] .01131 .32273 1 0 Mshowa p .002 w .02381 .41626 m .03006 .41626 L s P [(-5)] .01131 .41626 1 0 Mshowa p .002 w .02381 .50979 m .03006 .50979 L s P [(-2.5)] .01131 .50979 1 0 Mshowa p .001 w .02381 .06085 m .02756 .06085 L s P p .001 w .02381 .07955 m .02756 .07955 L s P p .001 w .02381 .09826 m .02756 .09826 L s P p .001 w .02381 .11697 m .02756 .11697 L s P p .001 w .02381 .15438 m .02756 .15438 L s P p .001 w .02381 .17308 m .02756 .17308 L s P p .001 w .02381 .19179 m .02756 .19179 L s P p .001 w .02381 .21049 m .02756 .21049 L s P p .001 w .02381 .24791 m .02756 .24791 L s P p .001 w .02381 .26661 m .02756 .26661 L s P p .001 w .02381 .28532 m .02756 .28532 L s P p .001 w .02381 .30402 m .02756 .30402 L s P p .001 w .02381 .34144 m .02756 .34144 L s P p .001 w .02381 .36014 m .02756 .36014 L s P p .001 w .02381 .37885 m .02756 .37885 L s P p .001 w .02381 .39755 m .02756 .39755 L s P p .001 w .02381 .43497 m .02756 .43497 L s P p .001 w .02381 .45367 m .02756 .45367 L s P p .001 w .02381 .47238 m .02756 .47238 L s P p .001 w .02381 .49108 m .02756 .49108 L s P p .001 w .02381 .5285 m .02756 .5285 L s P p .001 w .02381 .5472 m .02756 .5472 L s P p .001 w .02381 .56591 m .02756 .56591 L s P p .001 w .02381 .58461 m .02756 .58461 L s P p .001 w .02381 .02344 m .02756 .02344 L s P p .001 w .02381 .00473 m .02756 .00473 L s P p .002 w .02381 0 m .02381 .61803 L s P P p p .004 w .02381 .60332 m .06349 .50045 L .10317 .40097 L .14286 .30812 L .18254 .22483 L .22222 .15362 L .2619 .09639 L .28175 .07342 L .30159 .05432 L .32143 .03909 L .33135 .03292 L .34127 .02769 L .35119 .02338 L .36111 .01998 L .37103 .01745 L .37599 .0165 L .38095 .01576 L .38343 .01546 L .38591 .01522 L .38839 .01502 L .38963 .01494 L .39087 .01488 L .39211 .01482 L .39335 .01478 L .39459 .01474 L .39583 .01472 L .39707 .01472 L .39831 .01472 L .39955 .01473 L .40079 .01476 L .40203 .01479 L .40327 .01484 L .40575 .01497 L .40823 .01514 L .41071 .01535 L .41567 .0159 L .42063 .01661 L .43056 .01846 L .44048 .02086 L .46032 .02698 L .5 .04214 L .53968 .05773 L .57937 .07332 L .61905 .08891 L .65873 .10449 L .69841 .12008 L .7381 .13567 L .77778 .15126 L .81746 .16685 L .85714 .18244 L Mistroke .89683 .19802 L .93651 .21361 L .97619 .2292 L Mfstroke P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; pictureLeft = 141; pictureWidth = 248; pictureHeight = 153; endGroup] We come now to the final test. Compute and plot the n-Laplacian of j[r,a]. We use here the fact that j[r,a] is radial and that if first derivative with respect to r does not vanish for r>0. :[font = input; preserveAspect; startGroup] nlapwww[r_,a_,n_]=D[r^{n-1} ( (-D[www[r,a],r])^{n-2} D[www[r,a],r]),r]; :[font = input; preserveAspect] nlapv[r_,a_,n_]=D[r^{n-1} ( (-D[v[r,a],r])^{n-2} D[v[r,a],r]),r]; :[font = input; preserveAspect; startGroup] nlapu[r_,a_,n_]=D[r^{n-1} ( (-D[u[r],r])^{n-2} D[u[r],r]),r]//Simplify :[font = output; output; inactive; preserveAspect; endGroup] {0} ;[o] {0} :[font = text; inactive; preserveAspect] (...just kidding...) :[font = input; preserveAspect] nlapj[r_,a_,n_]:=Which[ r<=a/2,nlapwww[r,a,n], a/2a,nlapu[r,a,n] ] :[font = text; inactive; preserveAspect] Here comes the interesting picture. We will plot the n-Laplacian of j[r,.1] in dimensions 3, 4 and 5. :[font = input; preserveAspect; startGroup] Plot[{nlapj[r,.1,3],nlapj[r,.1,4],nlapj[r,.1,5]}, {r,0.0001,.12},PlotStyle->Thickness[.006]] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 20; pictureWidth = 512; pictureHeight = 316; endGroup; endGroup] %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 7.936508 0.603319 0.029146 [ [(0.02)] .18254 .60332 0 2 Msboxa [(0.04)] .34127 .60332 0 2 Msboxa [(0.06)] .5 .60332 0 2 Msboxa [(0.08)] .65873 .60332 0 2 Msboxa [(0.1)] .81746 .60332 0 2 Msboxa [(0.12)] .97619 .60332 0 2 Msboxa [(-20)] .01131 .02039 1 0 Msboxa [(-15)] .01131 .16612 1 0 Msboxa [(-10)] .01131 .31186 1 0 Msboxa [(-5)] .01131 .45759 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .18254 .60332 m .18254 .60957 L s P [(0.02)] .18254 .60332 0 2 Mshowa p .002 w .34127 .60332 m .34127 .60957 L s P [(0.04)] .34127 .60332 0 2 Mshowa p .002 w .5 .60332 m .5 .60957 L s P [(0.06)] .5 .60332 0 2 Mshowa p .002 w .65873 .60332 m .65873 .60957 L s P [(0.08)] .65873 .60332 0 2 Mshowa p .002 w .81746 .60332 m .81746 .60957 L s P [(0.1)] .81746 .60332 0 2 Mshowa p .002 w .97619 .60332 m .97619 .60957 L s P [(0.12)] .97619 .60332 0 2 Mshowa p .001 w .05556 .60332 m .05556 .60707 L s P p .001 w .0873 .60332 m .0873 .60707 L s P p .001 w .11905 .60332 m .11905 .60707 L s P p .001 w .15079 .60332 m .15079 .60707 L s P p .001 w .21429 .60332 m .21429 .60707 L s P p .001 w .24603 .60332 m .24603 .60707 L s P p .001 w .27778 .60332 m .27778 .60707 L s P p .001 w .30952 .60332 m .30952 .60707 L s P p .001 w .37302 .60332 m .37302 .60707 L s P p .001 w .40476 .60332 m .40476 .60707 L s P p .001 w .43651 .60332 m .43651 .60707 L s P p .001 w .46825 .60332 m .46825 .60707 L s P p .001 w .53175 .60332 m .53175 .60707 L s P p .001 w .56349 .60332 m .56349 .60707 L s P p .001 w .59524 .60332 m .59524 .60707 L s P p .001 w .62698 .60332 m .62698 .60707 L s P p .001 w .69048 .60332 m .69048 .60707 L s P p .001 w .72222 .60332 m .72222 .60707 L s P p .001 w .75397 .60332 m .75397 .60707 L s P p .001 w .78571 .60332 m .78571 .60707 L s P p .001 w .84921 .60332 m .84921 .60707 L s P p .001 w .88095 .60332 m .88095 .60707 L s P p .001 w .9127 .60332 m .9127 .60707 L s P p .001 w .94444 .60332 m .94444 .60707 L s P p .002 w 0 .60332 m 1 .60332 L s P p .002 w .02381 .02039 m .03006 .02039 L s P [(-20)] .01131 .02039 1 0 Mshowa p .002 w .02381 .16612 m .03006 .16612 L s P [(-15)] .01131 .16612 1 0 Mshowa p .002 w .02381 .31186 m .03006 .31186 L s P [(-10)] .01131 .31186 1 0 Mshowa p .002 w .02381 .45759 m .03006 .45759 L s P [(-5)] .01131 .45759 1 0 Mshowa p .001 w .02381 .04954 m .02756 .04954 L s P p .001 w .02381 .07869 m .02756 .07869 L s P p .001 w .02381 .10783 m .02756 .10783 L s P p .001 w .02381 .13698 m .02756 .13698 L s P p .001 w .02381 .19527 m .02756 .19527 L s P p .001 w .02381 .22442 m .02756 .22442 L s P p .001 w .02381 .25356 m .02756 .25356 L s P p .001 w .02381 .28271 m .02756 .28271 L s P p .001 w .02381 .341 m .02756 .341 L s P p .001 w .02381 .37015 m .02756 .37015 L s P p .001 w .02381 .39929 m .02756 .39929 L s P p .001 w .02381 .42844 m .02756 .42844 L s P p .001 w .02381 .48673 m .02756 .48673 L s P p .001 w .02381 .51588 m .02756 .51588 L s P p .001 w .02381 .54503 m .02756 .54503 L s P p .001 w .02381 .57417 m .02756 .57417 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p p .006 w .0246 .60332 m .02584 .60332 L .02708 .60332 L .02832 .60331 L .02956 .6033 L .0308 .60328 L .03204 .60326 L .03328 .60323 L .03452 .60319 L .03575 .60314 L .03699 .60308 L .03947 .60293 L .04195 .60271 L .04443 .60242 L .04938 .60162 L .05186 .60108 L .05434 .60044 L .0593 .59882 L .06425 .59669 L .06921 .59401 L .07417 .5907 L .08408 .58205 L .09399 .57039 L .1039 .55549 L .12373 .51546 L .14355 .46212 L .1832 .3247 L .22285 .17589 L .24268 .11124 L .25259 .08379 L .2625 .06039 L .27241 .0415 L .28232 .02749 L .28728 .02239 L .28976 .02033 L .29224 .01858 L .29472 .01717 L .29719 .01607 L .29843 .01564 L .29967 .0153 L .30091 .01503 L .30215 .01485 L .30339 .01474 L .30463 .01472 L .30587 .01477 L .30711 .0149 L .30834 .01511 L .30958 .01539 L .31082 .01576 L .31206 .01619 L Mistroke .31454 .0173 L .31702 .0187 L .32197 .02236 L .32693 .02713 L .33189 .03292 L .3418 .04724 L .36162 .0838 L .38145 .1236 L .39136 .14134 L .40127 .15553 L .40623 .16075 L .40871 .16279 L .41119 .1644 L .41242 .16504 L .41366 .16556 L .4149 .16596 L .41614 .16624 L .41738 .16639 L .41862 .1664 L .41986 .16627 L .4211 .1663 L .42358 .16723 L .42605 .1682 L .43101 .17023 L .44092 .1747 L .46075 .18517 L .5004 .2119 L .54005 .2456 L .5797 .2854 L .61935 .33043 L .65899 .37982 L .69864 .43269 L .73829 .48818 L .77794 .54541 L .79777 .57441 L .80768 .58895 L .81264 .59623 L .81511 .59987 L .81635 .60169 L .81759 .60332 L .82007 .60332 L .82255 .60332 L .8275 .60332 L .83742 .60332 L .85724 .60332 L .89689 .60332 L .93654 .60332 L .97619 .60332 L Mfstroke P P p p .006 w .0246 .60332 m .02584 .60332 L .02708 .60332 L .02832 .60332 L .02956 .60332 L .0308 .60332 L .03204 .60332 L .03328 .60332 L .03452 .60332 L .03575 .60332 L .03699 .60332 L .03823 .60332 L .03947 .60332 L .04071 .60332 L .04195 .60332 L .04319 .60331 L .04443 .60331 L .04567 .60331 L .04691 .60331 L .04938 .6033 L .05062 .6033 L .05186 .60329 L .05434 .60328 L .05682 .60326 L .0593 .60323 L .06177 .60319 L .06425 .60315 L .06673 .60309 L .06921 .60302 L .07169 .60293 L .07417 .60282 L .07912 .60253 L .08408 .60212 L .08903 .60156 L .09399 .60082 L .09895 .59985 L .1039 .59863 L .10886 .59709 L .11381 .59519 L .12373 .59014 L .13364 .58307 L .14355 .57358 L .15346 .56129 L .16338 .54591 L .1832 .50508 L .20303 .45078 L .22285 .38504 L .2625 .23924 L .28232 .17324 L .29224 .14519 L Mistroke .30215 .12147 L .31206 .10266 L .31702 .0952 L .32197 .08908 L .32693 .0843 L .32941 .0824 L .33189 .08083 L .33436 .07959 L .3356 .07908 L .33684 .07866 L .33808 .07831 L .33932 .07804 L .34056 .07784 L .3418 .07772 L .34304 .07767 L .34428 .07769 L .34552 .07778 L .34676 .07793 L .34923 .07845 L .35047 .07881 L .35171 .07922 L .35667 .08146 L .36162 .08453 L .37154 .09249 L .38145 .10166 L .3864 .10617 L .39136 .11029 L .39632 .11378 L .3988 .11519 L .40127 .11634 L .40251 .11679 L .40375 .11717 L .40499 .11746 L .40623 .11766 L .40747 .11777 L .40871 .11777 L .40995 .11766 L .41119 .11744 L .41242 .11711 L .41366 .11664 L .4149 .11605 L .41614 .11533 L .41862 .11345 L .41986 .11229 L .4211 .11129 L .42358 .1103 L .42605 .10938 L .43101 .10771 L .43597 .10629 L .44092 .10511 L Mistroke .44588 .10417 L .44836 .10379 L .45083 .10347 L .45331 .10321 L .45455 .10311 L .45579 .10302 L .45703 .10294 L .45827 .10288 L .45951 .10283 L .46075 .10281 L .46199 .10279 L .46323 .10279 L .46446 .10281 L .4657 .10284 L .46694 .10289 L .46818 .10295 L .47066 .10312 L .47314 .10335 L .47562 .10364 L .48057 .10441 L .48553 .10541 L .49048 .10667 L .5004 .10989 L .51031 .11408 L .52022 .11921 L .54005 .13229 L .5797 .16924 L .61935 .21949 L .65899 .28143 L .69864 .35312 L .73829 .43233 L .77794 .51668 L .79777 .55998 L .80768 .58178 L .81264 .59269 L .81511 .59815 L .81635 .60088 L .81759 .60332 L .82007 .60332 L .82255 .60332 L .8275 .60332 L .83742 .60332 L .85724 .60332 L .89689 .60332 L .93654 .60332 L .97619 .60332 L Mfstroke P P p p .006 w .0246 .60332 m .02584 .60332 L .02708 .60332 L .02832 .60332 L .02956 .60332 L .0308 .60332 L .03204 .60332 L .03328 .60332 L .03452 .60332 L .03575 .60332 L .03699 .60332 L .03823 .60332 L .03947 .60332 L .04071 .60332 L .04195 .60332 L .04319 .60332 L .04443 .60332 L .04567 .60332 L .04691 .60332 L .04815 .60332 L .04938 .60332 L .05062 .60332 L .05186 .60332 L .0531 .60332 L .05434 .60332 L .05682 .60332 L .05806 .60332 L .0593 .60332 L .06054 .60332 L .06177 .60332 L .06301 .60332 L .06425 .60332 L .06673 .60331 L .06921 .60331 L .07169 .60331 L .07417 .6033 L .07664 .60329 L .07912 .60329 L .0816 .60327 L .08408 .60326 L .08656 .60324 L .08903 .60322 L .09151 .60319 L .09399 .60315 L .09895 .60305 L .10142 .60299 L .1039 .60291 L .10886 .60271 L .11381 .60243 L .11877 .60206 L Mistroke .12373 .60156 L .12868 .60092 L .13364 .60009 L .1386 .59905 L .14355 .59775 L .15346 .59421 L .15842 .59187 L .16338 .5891 L .17329 .58201 L .1832 .57253 L .19311 .56029 L .20303 .54495 L .22285 .50423 L .24268 .45039 L .2625 .3863 L .30215 .25262 L .31206 .22382 L .32197 .19871 L .33189 .1779 L .3418 .16173 L .34676 .15538 L .35171 .15016 L .35667 .146 L .36162 .14281 L .3641 .14155 L .36658 .14048 L .36906 .13958 L .37154 .13885 L .37649 .13776 L .37897 .13736 L .38145 .13702 L .3864 .13639 L .38888 .13604 L .39136 .13563 L .39384 .13511 L .39632 .13445 L .3988 .13361 L .40127 .13255 L .40623 .1296 L .40871 .12761 L .41119 .12522 L .41614 .11903 L .4211 .1109 L .43101 .09919 L .44092 .08856 L .46075 .07078 L .47066 .06373 L .48057 .05796 L .48553 .05557 L .49048 .05351 L Mistroke .49544 .05179 L .5004 .05042 L .50287 .04986 L .50535 .04939 L .50783 .049 L .51031 .0487 L .51155 .04858 L .51279 .04849 L .51403 .04842 L .51527 .04837 L .5165 .04834 L .51774 .04833 L .51898 .04835 L .52022 .04838 L .52146 .04844 L .5227 .04853 L .52394 .04863 L .52518 .04876 L .52766 .04908 L .53013 .04949 L .53509 .05057 L .54005 .05202 L .54996 .05598 L .55987 .06139 L .5797 .07649 L .59952 .09722 L .61935 .12344 L .65899 .19125 L .69864 .27719 L .73829 .37761 L .77794 .48808 L .79777 .54557 L .80768 .5746 L .81264 .58915 L .81511 .59643 L .81635 .60007 L .81759 .60332 L .82007 .60332 L .82255 .60332 L .8275 .60332 L .83742 .60332 L .85724 .60332 L .89689 .60332 L .93654 .60332 L .97619 .60332 L Mfstroke P P P % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; rightWrapOffset = 450; endGroup] Negative! Good! For a classical proof of these properties of our test function see the paper quoted above. ^*)