! " $#&% '() +*-,/.01 % 2.3
4 1536.7 '8.9:;9<
=?>A@CBEDGFIH9=KJLDNMOHPHEQR>
S QUTRDNVWXDB+Y[ZRZ;Y
dkrvbl]X~sm9j pEbl] pEg{em9{edsa]lp0]`_~s j}obvgi|Ga)r`2deb`psry\X{e~s
x
P]lpEbv_ y_ipEblatuEgdem~sx a~sm
j0~]`_iacj~]`dex)~s{Ideb`dkuE_6]`b j~smia ]`_debX]`_iacb`debX|GpEb`b`dkh{ka ~sj jpEbl] usrv~]la)n g{Inqpsr ~s{e{I]`_iahGac~sgi]`dkng{
~smgbla)ng{Uj~]`_iacj0~]`dex)bA]`_~]L_~ sa{kac~r iacm mgir deiu]`_iat{e~sbl]y]`_ir`a)a:}sac~r bf]`_ir`pEguE__debfde'g
acx a _~csa ~s{eb`p usr`ac~]`{k}9~||r`acx)de~]lacmK_deb C~rvj ~sm 2dem<|Ga)rvb`pE~s{edk]} _dexv_+j0~ sacb _dej ~
]lrvg{k}0usr`ac~]yz(_w b`gi|Ga)r`deblpsr
pEg{em5~s{eblp{edsa ]lp]`_~s j}nqac{e{kpqaz(_w bl]`gmacP]`bEg{edkaC{eg]l]la)r`hgx zR~pE{e~z[p"!#!cd
~smXdex w gusa)}:n psrI]`_iacdkrRnqrvdkacmb`_dk| ~smx pEj |~s6}:mgir deiu j} ]`_ir`a)aL}sac~rvbR~]R]`_iaL\Xgbl]lr ~s{ede~s
X~]`dkpE~s{(Xdksa)rvbvdk]}
_~s b:~r`a0~s{eblp<mga]lp<j}] pz(_w1~smideblpsrvb \ymir`a %$X~sb`blac{e{ ~sm
& ac\ymir`a bX~sm9]`_iar`acbl] psn;]`_ia'9~]`_iacj~]`dex)~s{)(x)dkacx acb b`]`dk]`gi]la *9|~rv]`dex)g{e~rf]`_~s
]`_ia:~smjdedebl]lr ~]`dksatbl]`~,++npsry~s{e{U]`_iacdkry_ac{k|
de~s{e{k}" pEg{emo{edsa ]lp]`_~s j} j ps]`_a)rC~smn~]`_ia)rA~sm5j }hr`ps]`_ia)rLnpsrC]`_iacdkrC|~]`dkacx a
~sm0b`gi||Gpsr`] R\Xm0{e~sb`] hg](ips] {kac~sb`]#]`_~s j}miac~rL\ dspnqpsr ~s{e{N]`_iaX_~||deacb`b(b`_a uE~csa
Erratum
Listed below are the four papers that have been produced from this thesis.
Chapter 2:
Axelsson, A.
Transmission problems and boundary operator algebras.
Integral Equations Operator Theory 50 (2004), no. 2, 147-164.
Chapter 3:
Axelsson, A.
Oblique and normal transmission problems for Dirac operators with strongly Lipschitz
interfaces.
Comm. Partial Differential Equations 28 (2003), no. 11-12, 1911-1941.
Chapter 4:
Axelsson, A.; McIntosh, A.
Hodge decompositions on weakly Lipschitz domains.
In: T. Qian, T. Hempfling, A. McIntosh, F. Sommen (eds.), Advances in Analysis and
Geometry, New Developments Using Clifford Algebras, ISBN 3-7643-6661-3,
Series Trends in Mathematics, Birkhauser Basel, 2004.
Chapter 5:
Axelsson, A.
_a ~sdej psn;]`_debX]`_iacbvdeb deby]lpouEdksa ~5j~]`_iacj0~]`dex)~s{Inrv~sja psr npsr b`x)~]l]la)rvdeu psn;ac{kacx ]lr`pEj~u
ia)]`dex C~csacbth6}<r`pEgiuE_b`gir`n~sx acb +a|r`p sa]`_~] ]`_ia'9~ Lac{e{R]lrv~sbvjdeb`b`dkpE9|rvpsh{kacj dk]`_+~
Lac~ 2{k} dk|bvxv_dk]! de6]la)r`nq~sx a"de'dk]laXaca)r`us}ipsrvj0b#6deb Lac{e{N|NpEb`acm5de
rvacm_ipE{ej1blacblaXnqpsrLrvac~s{
nqr`a2giacx)dkacb gr`]`_ia)rvjpsr`a" La uEdksa |r`acx)deb`a x pEmdk]`dkpEbApE5]`_ia j~]la)r de~s{Ux pEbl]`~s6]`b +~sm
~smo]`_a:nrva6giacx } _aco]`_debf]lrv~sb`jdeb`bvdkpEo|r`psh{kacjdeb ac{e{U|GpEblacm
p blpE{ksa]`_ia'9~ Lac{e{]lrv~sbvjdeb`b`dkpE<|r`psh'{kacj aacjhGacm <~ ac{e{b:a6g~]`dkpEb:de~sac{.
{edk|]`dex w dkrv~sxoa2g~]`dkpE+amia)sac{kps| ~9a hNpEgm~r`}Kde6]la)usrv~s{fja)]`_ip2m$]lpKblpE{ksa ]`_a wXdkr ~sx
]lrv~sb`jdeb`bvdkpE<|r`psh{kacj
_debtj a)]`_ipmgblacb ~ hGpEgm~r`}de6]la)usrv~s{ps|Ga)rv~]lpsr []`_ia
02 /
& 0 _dexv_ n~sx ]lpsrvdeblacb ]`_a mipEgih'{ka {e~c}sa)r |Gps]lac6]`de~s{Ips|Na)r ~]lpsr +a|r`p sabl|Gacx ]lrv~s{ acbl]`dej~]lacb
nqpsr:]`_deb rvps]`~]`dkpEKps|Ga)rv~]lpsr deNdk]laacia)r`us}+ipsrvj0b gb`deiu $yp2musa miacx pEj |GpEb`dk]`dkpEb:pE ac~ {k}
[dk|b`xv_dk]!tmipEj~sdeb
pKacbvgir`ao]`_~]blpE{eg]`dkpEb ]lpK]`_aw dkrv~sx ]lrv~sbvjdeb`b`dkpE |r`psh{kacj dema)acm blpE{ksa 9~ ac{e{b
a2g~]`dkpEb# La deP]lrvp2mgx a ~s
5& 5 +* 2 $+ /.& 0
~sx ]`deiu deo]`_ia ]lrv~sx a:bl|'~sx a
& }<bv_ip deu5]`_~]X]`_deb ps|Na)r ~]lpsr:x pEjjgi]lacb dk]`_9]`_ia]Lp h~sb`dexr`a 'acx ]`dkpEps|Na)r ~]lpsrvb#La ~r`a
~h{kat]lp |r`p sa:]`_~]f]`_ia 9~ ac{e{[]lrv~sb`j0deb`b`dkpEo|rvpsh{kacj deb Lac{e{U|GpEblacm
+a ~s{eblp |r`p sa Lac{e{. |NpEb`acmiacb`b npsrL~ x){e~sbvb(psnUpsh{ed2giaXwXdkr ~sx ]lrv~sb`jdeb`bvdkpE|r`psh{kacj0b dk]`_5~
bl]lrvpEiuE{k}[dk|b`xv_dk]!:deP]la)r`n~sx a"de5]`_ia Xbl|~sx a pE5]`_ia de6]la)r`n~sx a
_debCdebAb`_p 5h2}acj |'{kp }deiu
]`_ia! ac{e{edexv_<]lacxv_d6gia" _dexv_ uEdksacby~siuEg{e~rybl|Gacx ]lrv~s{Gacbl]`dej0~]lacbfpE]`_ia r`ps]`~]`dkpE<ps|Na)r ~]lpsr
_debC]`_iacb`deb dex){egmiacbA|~rv]`bApsn[]`_a:npE{e{kp deiu0|~|Ga)rvb
\2ac{ebvblpEP\ "#XrvpsuE~rvm$! " $ypsuE~s "
x
6]lpEb`_P\ "$y~rvjpEdexC~s~s{k}b`debRpsnNw dkrv~sxCps|Ga)rv~]lpsr b
pE% dk|'b`xv_dk]!KmpEj~sdeb %C{ed.+Gpsrvm~s~s{k}b`deb ~smdk]`b ~||'{edex)~]`dkpEb z;rv~uEga"'&)(*(*( +&*,-/.0&2143
X\ 5 (x)d (2a)r <~]`_ (z(_6}2b C_iacj 6&*78:{eg La)r \Xx)~sm(z gih{ wXpsrvmir`acx _P]#9&)(*(:-
_ia:j~sdeor`acbvg{k]fuEdksacdez;r`ps|GpEb`dk]`dkpE;3 <- =7
\2ac{ebvblpE6\ 5 h{ed6ga ~sm ipsr j~s{i]lrv~sb`jdeb`bvdkpE|r`psh{kacj0bn psrLw dkrv~sxfps|Ga)rv~]lpsrvb dk]`_0bl]lr`pEiuE{k}
[dk|b`xv_dk]!tdeP]la)rvnq~sx acb
p0~||Gac~r de LpEj0j RzI~rv]`de~s{ w d.+Na)rvacP]`de~s{?> 2g~]`dkpEb@&)(*($&
\ sa)rvbvdkpEdex){egmacm ~sb C_~|]la)r,
\2ac{ebvblpE\
rv~sb`j0deb`b`dkpE |rvpsh{kacjbA~sm hNpEgm~r`} ps|Ga)rv~]lpsr ~s{kusa)hrv~sb
p0~||Gac~r de 6]la
usrv~s{A> 2g~]`dkpEb
5
|Ga)rv~]lpsr
_ia)psr`}&)(*($&
!" $#
% &
'(*) +' ,
, -.(/010 23 4
- <- C{ed.+Gpsrvm ~s{kusa)hrv~ 5
- =& C~s{ex)g{egb dk]`_76 - (
- =,
_r`a)a:x){e~sb`b`dex)~s{[a6g~]`dkpEb -7
- =,
_ia 8 a2g~]`dkpE -7
- =, =&
_ia ~|{e~sx a a6g~]`dkpE -3
- =, =, 9~ ac{e{bya6g~]`dkpEb -(5
- 1
5
|Ga)rv~]lpsr ]`_ia)psr`} -:9
- =7 [dk|b`xv_dk]!tdeP]la)rvnq~sx acb &)(
; < 2(1:'>=?*@ 2AB)? 2:CD=BE( /F 2E ! G %
& <- & ~s~sx _<~s{kusa)hr ~sbfusacia)rv~]lacmh6}o] p rva acx ]`dkpEb
,-& =&
Lp h~sbvdex ]lrv~sb`j0deb`b`dkpE |rvpsh{kacjb ,+5
& =&
_ia 8 a2g~]`dkpE ,+5
& =& =&
_ia ~|{e~sx a a6g~]`dkpE 1$(
G HI*JK) L 2M'N O( (P:Q=*SRTVUW0( !X=Y /((A$Z
:( [$CD\]0=:Z?$^W0(RT 2+3 _4
, <- (2|Gacx ]lrv~s{Gacb`]`dej~]lacbfnqpsr psh'{ed6gia:w dkrv~sxt]lrv~sb`j0deb`b`dkpEo|rvpsh{kacjb 1a`
, =&
_aLipsr j~s{E]lr ~sb`jdebvb`dkpEt|r`psh{kacj ~sm]`_iaLa2]la)rvdkpsr deP]la)rvdkpsr;ma)rvdk ~]`dksaAps|Ga)rv~]lpsr7*7
_ b77?+1=B2$@>c 2K$Cd\]0=:Z?$^ON 0 4[e
1 <- Xde{k|Gps]lacP]yps|Na)r ~]lpsrvb 3"`
1 =& $yp2musatmiacx pEj |GpEb`dk]`dkpEb f5),
g <
2(1:'=?*hRTiUW0 !/jkl 2PmX /#n/ojpqJK) !'q$Zrn 2K$C
\s0=:'Z*$(^W0(ERT !+ tK,
7
_a acia)r`us}o]lrv~sx atbl|'~sx ahu v`*&
4 [N +1=?ENE( 2(C>(()*$( , % G
3 <- w g~s{edk]O} de ]lrv~sx abl|~sx acb - ($,
3 =& $ypmiusatmiacx pEj|NpEbvdk]`dkpE psnR]lrv~sx atbl|~sx acb - ($3
-.")?(@R (=B" , %
0*2( 2=?ZC ,2,
_aCjps]`dk ~]`dkpEnpsr(]`_deb]`_iacb`deb x pEj acbnqr`pEj ]`_iaybl]`gmi} psnGbvx)~]l]la)rvdeiu:psnGac{kacx ]lr`pEj0~uEia)]`dex C~csacb
h2}5~ r`pEgiuE_ b`gr`nq~sx a +ax pEb`demia)r ]`_ia:nqpE{e{kp deiu <~ ac{e{[]lrv~sbvjdeb`b`dkpEo|r`psh{kacj
a)] hGa5~<b`gr`nq~sx a de6]la)r`n~sx a bla)|~rv~]`deiu9]`_ia5hGpEgmiacmKde6]la)rvdkpsrmipEj0~sde~sm
]`_iaa2]la)rvdkpsrmipEj~sde (gi||GpEbla]`_~] ac~sxv_ psnL]`_amipEj~sdeb
~sm deb x pEj |GpEblacm+psn
~5{edeiac~r _ipEjpsusacia)pEgb#Gdeblps]lr`ps|'dex U|NpEbvb`dkh{k} x pEmgx ]`deiu j~]la)rvde~s{ dk]`_9|Ga)rvjdk]l]`dkdk]} "!$# (
|Ga)rvjac~hde{edk]} %!&# (+~sm/x pEmgx ]`dkdk]} '!( ( /wXa 'ia; "!)*,+ -! .0/'!21 y_ia)r`a&deb
]`_iaonqr`a2giacx } ! jpEipxv_ir`pEj0~]`dex (d a ]`dej a _~rvjpEdex ;ac{kacx ]lrvdex~sm/j~uEia)]`dexac{emb
3 54%687 9+ 3 54 ;:<>=@?BAL~smDC 54%687 9+C 54 ;:E>=,?FA(b`~]`deblnq} 9~ ac{e{bXa2g~]`dkpEb
GIH CJ+%(K6
L
/MC0. GND3 +%(K6
L / !) 3 L GN PO QER C S+%(K6 GIH !) 3 9+%(KT
# dksac+~sdex pEj0deiuac{emU 3WV 6-C V<X deYZ 9'm<~0]lrv~sb`j0dk]l]lacm ac{emU 3 6-C X de[ ~sm~
r`a 'acx ]lacm ac{emU
3 26-C\ X de]/b`gxv_]`_~]^U 3 !_6-C! X
b`~]`deb acb 9~ ac{e{bta6g~]`dkpEb:de]!
U 3 `6-Ca X bv~]`debacb ]`_ia(de{ksa)r . cbg{e{ka)rLrv~smde~]`dkpE5x pEmdk]`dkpE5~]ed ~smpED La _~ say]`_a2f gj |
r`ac{e~]`dkpEb g H hC V .C L C 9+%(K6 g
N 3
V . 3 L 3 9+%(K6 g N PO Qji hC V .C L O Q<k C
9+%(K6 -,
g H ) 3 V . 3 L ) 3 9+%(K6 _a)r`a g
miacips]lacbf]`_ia:pEgi] L~rvm|GpEdeP]`deiu gdk]fipsrvj0~s{GpEl
_a:j~sdeo]`_iacjacbAde ]`_debf]`_acb`debf~r`a:]`_a:npE{e{kp deiu
d>(jmacmmdeiupsn <~ ac{e{bya2g~]`dkpEb de~s ac{e{edk|]`dextw dkrv~sxta6g~]`dkpE
dedo\/|gir`a rvbl][psrvmia)rR~||r`pE~sx _ ]lp w dkrv~sx)b~sm <~ Lac{e{ba6g~]`dkpEb ?+aCma)sac{kps| j a)]`_ipmb
npsrmdkr`acx ]`{k} Lpsrdeiu dk]`_ ]`_arvbl]psrvma)r wXdkr ~sx a6g~]`dkpEAdebl]lac~sm/psnXr`acmgx)deu+]lp ~
$yde{khGa)r`]ybl|~sx a demgx a:]`_iabl|{edk]l]`deiu + L\X{eblprvac{ka) ~s6] ]lp0]lrv~sb`j0deb`b`dkpE
|r`psh{kacj0bA~r`a$yp2musa ]}2|Na bl|{edk]l]`deiuEb+
)
)
demgx acmoh6}
~ de{k|Gps]lac6]fps|Ga)rv~]lpsr 'da
+%(
dk r`acm_pE{ejps|Ga)rv~]lpsrf]`_ia)psrv}
+a ip^uEdksa ~s p sa)r`2dka psnRpEgirf~||rvpE~sxv_ ]lp0blpE{k2deu ]`_ia9~ Lac{e{ ]lr ~sb`jdebvb`dkpEo|r`psh{kacj
\ gb`g~s{C~||rvpE~sxv_ ]lp <~ ac{e{b0a2g~]`dkpEbdeb]lp mdeb`x)~r m ]`_ia ] Lp # ~sgb`b mdksa)rvusacx a
a6g~]`dkpEby~sbyr`acmgm~s6] $ypLa)sa)r Gh6} pE{k}ogb`deiu ]`_ia
~rv~sm~ }~sm <~ ac{e{.\Xj |a)rva
a6g~]`dkpEb ]`_a:|~r`]`de~s{ md.+Na)r`ac6]`de~s{ b`}2bl]lacj deb ip0{kpEiusa)r ac{e{edk|]`dex ]`_debf]`_acb`deb Latb`_ip
de<ma)]`~sde{I_p ]lp5blpE{ksa]`_ia <~ Lac{e{[]lr ~sb`jdebvb`dkpE |rvpsh{kacj -, dk]`_<ac{e{edk|]`dexj a)]`_ipmb#
gb`deiu]`_ia # ~sgb`bfa2g~]`dkpEb de<~0~]`grv~s{C~c}
_ia:h~sb`dex demiac~5deb ]lp0r`a frvdk]la <~ ac{e{b
a6g~]`dkpEb ~sby~s ac{e{edk|]`dextw dkrv~sxta6g~]`dkpE
U *,+0: O 8 O . : 8 . : 8 .>: V + (KT
$ a)rva *,+
L
/
)
: V"!^3 . O
#
Q
C deb5]`_ia9nqg{e{ ac{kacx ]lr`pEj0~uEia)]`dexac{em _ia)r`a a+_~ sa
r`a)|{e~sx acm ]`_ia+j~uEa)]`dex ac{em ~hGp sa dk]`_ ]`_iax psr`r`acbl|GpEmdeiu $yp2musa9mg~s{ac{em C +
C O : ! : .C' : ! : O .C : O
! : ~sm _ia)r`aPU / : V 6 :
O
6 :E6 :
X ~r`a ]`_ia h~sbvdebAsacx ]lpsrvbAnqpsr
<de sp b dRbl|~sx a~sml*,+
)
$deby]`_a C~csa 2gjhGa)r +a frvdk]la ! npsr ]`_iaa2]la)r dkpsr
|r`pmgx ]A~sm
nqpsrf]`_ia C{ed.+Gpsrvm |r`p2mgx ]
O] deb dej |Gpsr`]`~s6]y]lpoips]la]`_~]*9~ Lac{e{b a6g~]`dkpEb x pEjhdeia]lpo~5w dkrv~sxa6g~]`dkpEUhgi]
~s{eblp]`_~]A]`_ia ac{kacx ]lrvpEj~uEia)]`dex''ac{em debf~bl|Gacx)de~s{UblpE{egi]`dkpE psn U$ + (~sbfdk]Adebf~|gir`a
hdksacx ]lpsr 'ac{em *M&% 6
(' (gir`|r deb`deiuE{k}";dk]]`girvbtpEgi]]`_~] La x)~s r`a)nqpsrvjg{e~]la
]`_deb x pEb`]lrv~sdeP]pE]`_ia+rv~siusa+psn ~sb ~/mdksa)r`usacx a x)girv{ nqr`a)aKx pEmdk]`dkpE8pE bla)a
z;r`ps|GpEb`dk]`dkpE, =& =7
p$blpE{ksa]`_a ]lrv~sb`j0deb`b`dkpE/|r`psh{kacj nqpsr]`_ia<wXdkr ~sx ps|Ga)rv~]lpsr U a<mia)sac{kps| ~?ia
hGpEgm~r`} deP]la)usrv~s{ j a)]`_ipm
psr<bvdej |{edex)dk]O}":x pEb`demia)r ]`_ia*) r`ac{e~]`dk2deb`]`dex,+ x)~sbla _iac
) + ) 9 o]`_debfx)~sbla" -,Lx)~s hGafr dk]l]lac de ]`_ia:nqpsrvj
-. . . / . . .0 1 32254 L
2 4 2+
1 76 pEl6 1 32 84' L 2 4 2+ 1 6 pEl6
U ![+%( de !MT
&
$ a)rva 6 debfmia)]la)r jdeiacmoh2} U
3 V 6-C
V X
!$debA]`_ia ac{kacx ]lr`pEj0~uEia)]`dex'ac{em dk]`_ x pEj |GpEiac6]`b
U 3 ! 6-C ! X ~sm9hGpEgm~r`}]lrv~sx a(4 ! U]`_iaZf gj |9|~rv~sj a)]la)r b"2 ! ~r`a mia)]la)rvj0deiacm9h6} !)
~sm ! ~sm
1
~sm
1
miacips]la<]`_ia ]`~siusac6]`de~s{ ~sm ipsr j~s{C|r`pEf acx ]`dkpE ps|Na)r ~]lpsrvb
r`acbl|Gacx ]`dksac{k} ) 0]`_debCwXdkr ~sxy]lr ~sb`jdebvb`dkpE|r`psh{kacj ! ~sm
6
de0usacia)rv~s{N]`~ sa ~s{egiacbCde
]`_ia:ng{e{Ga]la)rvdkpsry~s{kusa)hr ~6[+V699
O
\ybvblp2x)de~]lacm ydk]`_ ]`_iaw dkrv~sxtps|Ga)rv~]lpsr
U
deb ~ C~sgxv_6} ]}2|Ga b`deiuEg{e~rXde6]la)usrv~s{Ips|Ga)rv~]lpsr
]lrv~sx acbApsn[jpEipxv_ir`pEj0~]`dex'ac{embCde !+h2} 3
La j~ sa ]`_ia ~sb`~]! 4'!Y+ 3
4 _a)r`a
4I+ 4 . 4 +aip de6]lr`pmgx a]`_a<r`a 'acx ]`dkpE ps|Ga)rv~]lpsrvb
3 *,+ 3 L 3 ~sm
1 *,+ 1
L
1 _ia)rva 3
+ 1
+ ~sb Lac{e{U~sbA]`_ia 02 /. 0 3 1 Egbl]A~sb
pEia gblacb ]`_a mipEgih{ka {e~ }sa)r |Gps]lac6]`de~s{ps|Ga)rv~]lpsrt]lpb`pE{ksa ]`_ia w dkrvdexv_{ka)]thGpEgm~rv} ~s{ega
|r`psh{kacjnqpsrA]`_ia ~|{e~sx a ps|Ga)rv~]lpsr blp a:gb`aX]`_aXrvps]`~]`dkpE ps|Ga)rv~]lpsr
3 1
]lp blpE{ksat]`_ia
wXdkr ~sx ]lrv~sb`j0deb`b`dkpEo|rvpsh{kacj mia)acm & CdebAa6gdk ~s{kac6] ]lp ]`_atdeP]la)usr ~s{ a2g~]`dkpE
L
3 1 4 + & 3 6 6
_ia)r`a *,+ 32 . 2` ;132
L
2` \ r`acj0~r?_ia)rvadeb]`_~] ]`_a r`ps]`~]`dkpE ps|Na)r ~]lpsr
acb`blac6]`de~s{e{k}5n~sx ]lpsrvdeblacb ]`_ia mipEgih{ka {e~ }sa)r |Gps]lacP]`de~s{G]O}6|Ga ps|Ga)rv~]lpsr <flgbl]A~sb U b6g~rvacbC]lp
]`_ia$yac{ej_ipE{k]! ps|Ga)rv~]lpsr U
+ .
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